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<nowiki>{{</nowiki>
<!--{{about|infinite sums|finite sums|Summation}}-->
{{Kalkulus}}
Suatu deret dikatakan konvergen secara absolut jika deret yang terbentuk dari nilai absolut syarat pada konvergen; yaitu, diberi urutan tak terbatas <math>\{a_k\}</math>:
:<math>\sum_{k = 1}^\infty |a_k| \text{konvergensi}</math>
konvergensi.[[Berkas:Exp series.gif|ka|jmpl|[[:en:exponential function|Fungsi eksponensial]] (biru), dan jumlah ''n''+1 elemen pertama dari [[:en:Maclaurin series|deret pangkat Maclaurin]] (merah).]]
=== Teorema ===
== Deret pangkat ==
{{Artikel|Deret pangkat}}
'''Deret pangkat''' (satu variabel) dalam [[matematika]] adalah [[deret tak terhingga]] dalam bentuk
:<math>f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots</math>
dengan ''a<sub>n</sub>'' melambangkan koefisien suku ke-''n'', ''c'' adalah konstanta dan ''x'' berubah-ubah di sekitar ''c'' (karena alasan ini kadang-kadang deret seperti ini dikatakan ''berpusat'' di ''c''). Deret ini biasanya berupa [[deret Taylor]] dari suatu [[fungsi]].
Pada banyak keadaan ''c'' sama dengan nol, contohnya pada [[:en:Maclaurin series|deret Maclaurin]]. Dalam hal tersebut deret pangkat mengambil bentuk yang lebih sederhana:
::<math>
f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots.
</math>
Deret pangkat biasanya ditemukan dalam [[analisis matematika]], tetapi juga dapat ditemukan pada [[kombinatorika]] (dengan nama [[fungsi pembangkit]]), dan pada [[teknik elektro]] (dengan nama [[transformasi Z]]).
[[Berkas:Exp series.gif|ka|jmpl|[[:en:exponential function|Fungsi eksponensial]] (biru), dan jumlah ''n''+1 elemen pertama dari [[:en:Maclaurin series|deret pangkat Maclaurin]] (merah).]]
== Deret Fourier ==
'''Barisan jumlah parsial''' <math>\{S_k\}</math> bersangkutan dengan suatu deret <math>\sum_{n=0}^\infty a_n</math> didefinisikan bagi setiap <math>k</math> sebagai jumlah Barisan <math>\{a_n\}</math> dari <math>a_0</math> sampai <math>a_k</math>
:<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
Berdasarkan definisi, deret <math>\sum_{n=0}^{\infty} a_n</math> '''convergeskonvergen''' menjadi suatu limit <math>L</math> jika dan hanya jika urutan yang bersangkutan dengan jumlah parsial <math>\{S_k\}</math> [[Limit of a sequencebarisan#FormalDefinisi Definitionformal|converges]] menjadi <math>L</math>. Definisi ini biasanya ditulis sebagai
:<math>L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
<!--
More generally, if <math>I \xrightarrow{a} G</math> is a [[Function (mathematics)|function]] from an [[index set]] I to a set G, then the '''series''' associated to <math>a</math> is the [[formal sum]] of the elements <math>a(x) \in G </math> over the index elements <math>x \in I</math> denoted by the
:<math>\sum_{x \in I} a(x).</math>
When the index set is the natural numbers <math>I=\mathbb{N}</math>, the function <math>\mathbb{N} \xrightarrow{a} G</math> is a [[sequence]] denoted by <math>a(n)=a_n</math>. A series indexed on the natural numbers is an ordered formal sum and so we rewrite <math>\sum_{n \in \mathbb{N}}</math> as <math>\sum_{n=0}^{\infty}</math> in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
:<math>\sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots.</math>
When the set <math>G</math> is a [[semigroup]], the '''sequence of partial sums''' <math>\{S_k\} \subset G</math> associated to a sequence <math>\{a_n\} \subset G</math> is defined for each <math>k</math> as the sum of the terms <math>a_0,a_1,\cdots,a_k</math>
:<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
When the semigroup <math>G</math> is also a [[topological space]], then the series <math>\sum_{n=0}^{\infty} a_n</math> '''converges''' to an element <math>L \in G</math> if and only if the associated sequence of partial sums <math>\{S_k\}</math> [[Limit of a sequence#Formal Definition|converges]] to <math>L</math>. This definition is usually written as
:<math>L = \sum_{n=0}^{\infty} a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
=== Deret Konvergen ===
A series ∑''a<sub>n</sub>'' is said to ''''[[Convergent series|converge]]'''' or to 'be convergent' when the sequence ''S''<sub>''N''</sub> of partial sums has a finite [[Limit of a sequence|limit]]. If the limit of ''S''<sub>''N''</sub> is infinite or does not exist, the series is said to '''[[Divergent series|diverge]]'''. When the limit of partial sums exists, it is called the '''sum of the series'''
: <math>\sum_{n=0}^\infty a_n = \lim_{N\to\infty} S_N = \lim_{N\to\infty} \sum_{n=0}^N a_n.</math>
An easy way that an infinite series can converge is if all the ''a''<sub>''n''</sub> are zero for ''n'' sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series. Consider the example
:<math> 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.</math>
It is possible to "visualize" its convergence on the [[real number|real number line]]: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted ''S'', it can be seen that
:<math>S/2 = \frac{1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}{2} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16} +\cdots.</math>
Therefore,
:<math>S-S/2 = 1 \Rightarrow S = 2.\,\!</math>
Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a [[Repeating decimal|recurring decimal]], as in
:<math>x = 0.111\dots \, </math>
we are talking, in fact, just about the series
:<math>\sum_{n=1}^\infty \frac{1}{10^n}.</math>
But since these series always converge to [[real numbers]] (because of what is called the [[Complete space|completeness property]] of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and <sup>1</sup>/<sub>9</sub>. Less clear is the argument that {{nowrap|1=9 × 0.111… = 0.999… = 1}}, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See [[0.999...]] for more.
=== Contoh ===
* A ''[[geometric series]]'' is one where each successive term is produced by multiplying the previous term by a [[Mathematical constant|constant number]] (called the common ratio in this context). Example:
::<math>1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n}.</math>
:In general, the geometric series
::<math>\sum_{n=0}^\infty z^n</math>
:converges [[if and only if]] |''z''| < 1.
* An ''[[Arithmetico-geometric sequence]]'' is a generalization of the geometric series, which has coefficients of the common ratio equal to the terms in an [[arithmetic series]]. Example:
::<math>3 + {5 \over 2} + {7 \over 4} + {9 \over 8} + {11 \over 16} + \cdots=\sum_{n=0}^\infty{(3+2n) \over 2^n}.</math>
* The ''[[harmonic series (mathematics)|harmonic series]]'' is the series
::<math>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots = \sum_{n=1}^\infty {1 \over n}.</math>
:The harmonic series is [[harmonic series (mathematics)#Divergence|divergent]].
* An ''[[alternating series]]'' is a series where terms alternate signs. Example:
::<math>1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}=\ln(2).</math>
*The [[harmonic series (mathematics)#P-series|p-series]]
::<math>\sum_{n=1}^\infty\frac{1}{n^r}</math>
:converges if ''r'' > 1 and diverges for ''r'' ≤ 1, which can be shown with the integral criterion described below in [[Series (mathematics)#Convergence tests|convergence tests]]. As a function of ''r'', the sum of this series is [[Riemann zeta function|Riemann's zeta function]].
*A [[telescoping series]]
::<math>\sum_{n=1}^\infty (b_n-b_{n+1})</math>
:converges if the [[sequence]] ''b''<sub>''n''</sub> converges to a limit ''L'' as ''n'' goes to infinity. The value of the series is then ''b''<sub>1</sub> − ''L''.
===Calculus and partial summation as an operation on sequences===
Partial summation takes as input a sequence, { ''a''<sub>''n''</sub> }, and gives as output another sequence, { ''S''<sub>''N''</sub> }. It is thus a [[unary operation]] on sequences. Further, this function is [[linear function|linear]], and thus is a [[linear operator]] on the vector space of sequences, denoted Σ. The inverse operator is the [[finite difference]] operator, Δ. These behave as discrete analogs of [[integral|integration]] and [[derivative|differentiation]], only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that <math>\int_0^x 1\,dt = x.</math>
In [[computer science]] it is known as [[prefix sum]].
==Properties of series==
Series are classified not only by whether they converge or diverge, but also by the properties of the terms a<sub>n</sub> (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a<sub>n</sub> (whether it is a real number, arithmetic progression, trigonometric function); etc.
===Non-negative terms===
When ''a<sub>n</sub>'' is a non-negative real number for every ''n'', the sequence ''S<sub>N</sub>'' of partial sums is non-decreasing. It follows that a series ∑''a<sub>n</sub>'' with non-negative terms converges if and only if the sequence ''S<sub>N</sub>'' of partial sums is bounded.
For example, the series
:<math>\sum_{n \ge 1} \frac{1}{n^2}</math>
is convergent, because the inequality
:<math>\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, \quad n \ge 2,</math>
and a telescopic sum argument implies that the partial sums are bounded by 2.
===Absolute convergence===
{{Main|Absolute convergence}}
A series
:<math>\sum_{n=0}^\infty a_n</math>
is said to '''converge absolutely''' if the series of [[absolute value]]s
:<math>\sum_{n=0}^\infty \left|a_n\right|</math>
converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.
===Conditional convergence===
{{Main|Conditional convergence}}
A series of real or complex numbers is said to be '''conditionally convergent''' (or '''semi-convergent''') if it is convergent but not absolutely convergent. A famous example is the alternating series
:<math>\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots</math>
which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent [[Harmonic series (mathematics)|harmonic series]]. The [[Riemann series theorem]] says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the ''a''<sub>''n''</sub> are real and ''S'' is any real number, that one can find a reordering so that the reordered series converges with sum equal to ''S''.
[[Abel's test]] is an important tool for handling semi-convergent series. If a series has the form
:<math>\sum a_n = \sum \lambda_n b_n</math>
where the partial sums ''B''<sub>''N''</sub> = {{nowrap|''b''<sub>0</sub> + ··· + ''b<sub>n</sub>''}} are bounded, ''λ''<sub>''n''</sub> has bounded variation, and {{nowrap|lim λ<sub>''n''</sub> ''B''<sub>''n''</sub>}} exists:
:<math>\sup_N \Bigl| \sum_{n=0}^N b_n \Bigr| < \infty, \ \ \sum |\lambda_{n+1} - \lambda_n| < \infty\ \text{and} \ \lambda_n B_n \ \text{converges,}</math>
then the series {{nowrap|∑ ''a<sub>n</sub>''}} is convergent. This applies to the pointwise convergence of many trigonometric series, as in
:<math>\sum_{n=2}^\infty \frac{\sin(n x)}{\ln n}</math>
with 0 < ''x'' < 2π. Abel's method consists in writing ''b''<sub>''n''+1</sub> = ''B''<sub>''n''+1</sub> − ''B''<sub>''n''</sub>, and in performing a transformation similar to [[integration by parts]] (called [[summation by parts]]), that relates the given series {{nowrap|∑ ''a<sub>n</sub>''}} to the absolutely convergent series
:<math> \sum (\lambda_n - \lambda_{n+1}) \, B_n.</math>
==Convergence tests==
{{Main|Convergence tests}}
* ''[[n-th term test]]'': If lim<sub>''n''→∞</sub> ''a''<sub>''n''</sub> ≠ 0 then the series diverges.
*Comparison test 1 (see [[Direct comparison test]]): If ∑''b<sub>n</sub>'' is an [[absolute convergence|absolutely convergent]] series such that |''a<sub>n</sub>'' | ≤ ''C'' |''b<sub>n</sub>'' | for some number ''C'' and for sufficiently large ''n'' , then ∑''a<sub>n</sub>'' converges absolutely as well. If ∑|''b<sub>n</sub>'' | diverges, and |''a<sub>n</sub>'' | ≥ |''b<sub>n</sub>'' | for all sufficiently large ''n'' , then ∑''a<sub>n</sub>'' also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a<sub>n</sub>'' alternate in sign).
*Comparison test 2 (see [[Limit comparison test]]): If ∑''b<sub>n</sub>'' is an absolutely convergent series such that |''a<sub>n+1</sub>'' /''a<sub>n</sub>'' | ≤ |''b<sub>n+1</sub>'' /''b<sub>n</sub>'' | for sufficiently large ''n'' , then ∑''a<sub>n</sub>'' converges absolutely as well. If ∑|''b<sub>n</sub>'' | diverges, and |''a<sub>n+1</sub>'' /''a<sub>n</sub>'' | ≥ |''b<sub>n+1</sub>'' /''b<sub>n</sub>'' | for all sufficiently large ''n'' , then ∑''a<sub>n</sub>'' also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a<sub>n</sub>'' alternate in sign).
*[[Ratio test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''+1</sub>/''a''<sub>''n''</sub>|<''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
*[[Root test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''</sub>|<sup>1/''n''</sup> ≤ ''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely.
*[[Integral test for convergence|Integral test]]: if ''ƒ''(''x'') is a positive [[monotone decreasing]] function defined on the [[interval (mathematics)|interval]] <nowiki>[</nowiki>1, ∞<nowiki>)</nowiki>--><!--DO NOT "FIX" THE "TYPO" IN THE FOREGOING. IT IS INTENDED TO SAY [...) WITH A SQUARE BRACKET ON THE LEFT AND A ROUND BRACKET ON THE RIGHT. --><!-- with ''ƒ''(''n'') = ''a''<sub>''n''</sub> for all ''n'', then ∑''a''<sub>''n''</sub> converges if and only if the [[integral]] ∫<sub>1</sub><sup>∞</sup> ''ƒ''(''x'') d''x'' is finite.
*[[Cauchy's condensation test]]: If ''a''<sub>''n''</sub> is non-negative and non-increasing, then the two series ∑''a''<sub>''n''</sub> and ∑2<sup>''k''</sup>''a''<sub>(2<sup>''k''</sup>)</sub> are of the same nature: both convergent, or both divergent.
*[[Alternating series test]]: A series of the form ∑(−1)<sup>''n''</sup> ''a''<sub>''n''</sub> (with ''a''<sub>''n''</sub> ≥ 0) is called ''alternating''. Such a series converges if the [[sequence]] ''a''<sub>''n''</sub> is [[monotone decreasing]] and converges to 0. The converse is in general not true.
*For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]].
-->
== Deret fungsi ==
{{Main|Deret fungsi}}
:<math>\sum_{n=0}^\infty f_n(x)</math>
'''[[konvergen titik demi titiksesetitik]]''' pada suatu himpunan ''<math>E''</math>, jika deret itu ''convergeskonvergen'' untuk setiap ''<math>x''</math> dalamdi ''<math>E''</math> sebagai suatu deret ordinaribiasa dari bilangan real atau bilangan kompleks. EkuivalenJumlah denganparsialnya itu,ekuivalen jumlahdengan parsialdi atas.
:<math>s_N(x) = \sum_{n=0}^N f_n(x)</math>
Deret tersebut konvergen menjadike ''ƒ''<math>f(''x'')</math> sebagaiketika ''<math>N'' → ∞ \to \infty</math> untuk setiap ''<math>x'' ∈ '' \in E''</math>.
<!--
A stronger notion of convergence of a series of functions is called '''[[uniform convergence]]'''. The series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum,
:<math>|s_N(x) - f(x)|\ </math>
can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''.
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ''ƒ''<sub>''n''</sub> are [[integral|integrable]] on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the [[Weierstrass M-test|Weierstrass' M-test]], [[Abel's uniform convergence test]], [[Dini's test]]--><!--, and the [[Cauchy criterion]]: this is not about convergence of functions, even less about uniform convergence. -->.
<!--
More sophisticated types of convergence of a series of functions can also be defined. In [[measure theory]], for instance, a series of functions converges [[almost everywhere]] if it converges pointwise except on a certain set of [[null set|measure zero]]. Other [[modes of convergence]] depend on a different [[metric space]] structure on the space of functions under consideration. For instance, a series of functions '''converges in mean''' on a set ''E'' to a limit function ''ƒ'' provided
:<math>\int_E \left|s_N(x)-f(x)\right|^2\,dx \to 0</math>
as ''N'' → ∞.
-->
=== Deret pangkat ===
:{{Main|Deret pangkat}}
:
'''Deret pangkat''' adalah(satu suatuvariabel) dalam [[matematika]] adalah [[deret tak terhingga]] dalam bentuk
:<math>f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n. = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots</math>
dengan ''a<sub>n</sub>'' melambangkan koefisien suku ke-''n'', ''c'' adalah konstanta dan ''x'' berubah-ubah di sekitar ''c'' (karena alasan ini kadang-kadang deret seperti ini dikatakan ''berpusat'' di ''c''). Deret ini biasanya berupa [[deret Taylor]] dari suatu [[fungsi]].
Pada banyak keadaan ''c'' sama dengan nol, contohnya pada [[:en:Maclaurin series|deret Maclaurin]]. Dalam hal tersebut deret pangkat mengambil bentuk yang lebih sederhana:<math>
f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots.
</math>
Deret pangkat biasanya ditemukan dalam [[analisis matematika]], tetapi juga dapat ditemukan pada [[kombinatorika]] (dengan nama [[fungsi pembangkit]]), dan pada [[teknik elektro]] (dengan nama [[transformasi Z]]).
[[Deret Taylor]] pada suatu titik ''c'' pada suatu fungsi adalah suatu deret pangkat yang dalam banyak kasus berkonvergen menjadi suatu fungsi dalam lingkungan ''c''. Misalnya, deret
(in other words, ω<sub>1</sub> copies of 1 is ω<sub>1</sub>) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.
}}
-->:
=== Daftar suku dan jumlah suku ===
{| class="wikitable"
|-
! Nama bilangan !! Suku ke-n !! Jumlah suku ke-n
|-
| Bilangan asli || <math>n</math> || <math>\frac{n \cdot (n+1)}{2}</math>
|-
| Bilangan asli perkalian dua berurutan || <math>n \cdot (n+1)</math> || <math>\frac{n \cdot (n+1) \cdot (n+2)}{3}</math>
|-
| Bilangan asli perkalian tiga berurutan || <math>n \cdot (n+1) \cdot (n+2)</math> || <math>\frac{n \cdot (n+1) \cdot (n+2) \cdot (n+3)}{4}</math>
|-
| Bilangan asli perkalian k berurutan || <math>n \cdot (n+1) \cdot (n+2) \dots (n+k)</math> || <math>\frac{n \cdot (n+1) \cdot (n+2) \dots (n+k+1)}{k+1}</math>
|-
| Bilangan asli penyebut perkalian dua berurutan || <math>\frac{1}{n} - \frac{1}{(n+1)} = \frac{1}{n \cdot (n+1)}</math> || <math>\frac{n}{n+1}</math>
|-
| Bilangan ganjil || <math>2n-1</math> || <math>n^2</math>
|-
| Bilangan genap || <math>2n</math> || <math>n^2+n</math>
|-
| Bilangan Persegi/Kuadrat || <math>n^2</math> || <math>\frac{n \cdot (n+1) \cdot (2n+1)}{6}</math>
|-
| Bilangan Kubus/Kubik || <math>n^3</math> || <math>\frac{n^2 \cdot (n+1)^2}{4}</math>
|-
| Bilangan Segi tiga || <math>\frac{n \cdot (n+1)}{2}</math> || <math>\frac{n \cdot (n+1) \cdot (n+2)}{6}</math>
|-
| Bilangan Persegi panjang || <math>n \cdot (n+1)</math> || <math>\frac{n \cdot (n+1) \cdot (n+2)}{3}</math>
|-
| Bilangan Balok || <math>n \cdot (n+1) \cdot (n+2)</math> || <math>\frac{n \cdot (n+1) \cdot (n+2) \cdot (n+3)}{4}</math>
|}
== Contoh ==
=== [[Deret (matematika)#Notasi|Contoh 1]] ===
Contoh dari persamaan {{math|''f'' (''n'') {{=}} ''n''²}} Pada nilai produk {{math|''f'' (''n'')}} dari nilai {{math|''n''}} antara {{math|1}} dan {{math|∞}} dapat dinyatakan sebagai:
<math>\sum_{n = 1}^\infty n^2 = 1^2 + 2^2 + 3^2 +\cdots.</math>
Jika kita hanya menginginkan jumlah persyaratan hingga {{math|n {{=}} 1}} hal itu akan menjadi:
<math>S_{10} = \sum_{n = 1}^{10} n^2 = 1^2 + 2^2 + 3^2 +\cdots+ 10^2 = 385.</math>
=== [[Deret (matematika)#Konvergensi|Contoh 2]] ===
Contoh dari produk <math>\sum\limits_{n = 1}^\infty \frac{(-1)^{n+1}}{2^n}</math> menyatu secara mutlak. Tetapi <math>\sum\limits_{n = 1}^\infty \frac{(-1)^{n+1}}{n}
n=1</math> menyatu secara kondisional. Jika kita tahu bahwa keduanya konvergen, kita dapat membuktikan bahwa keduanya konvergen secara absolut atau bersyarat dengan mengambil jumlah nilai absolut dari fungsi tersebut:
<math>\begin{aligned} \sum\limits_{n = 1}^\infty \left|\frac{(-1)^{n+1}}{2^{n}}\right| &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^{n}} \rightarrow \text{konvergensi}\\ \sum\limits_{n = 1}^\infty \left|\frac{(-1)^{n+1}}{n}\right| &= 1 + \frac{1}{2} + \frac{1}{3} + \cdots +\frac{1}{n} \rightarrow \text{tidak konvergensi}. \end{aligned}</math>
=== Contoh 3 ===
Berapa nilai dari <math>(1 - \frac{1}{2}) \cdot (1 - \frac{1}{3}) \cdot (1 - \frac{1}{4}) \cdot (1 - \frac{1}{5})</math>?
: <math>(1 - \frac{1}{2}) \cdot (1 - \frac{1}{3}) \cdot (1 - \frac{1}{4}) \cdot (1 - \frac{1}{5}) = \frac{\mathbf{1}}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{\mathbf{5}} = \frac{1}{5}</math>
=== Contoh 4 ===
Berapa nilai dari <math>\frac{1}{4 \cdot 5} + \frac{1}{5 \cdot 6} + \frac{1}{6}</math>?
: <math>\frac{1}{4 \cdot 5} + \frac{1}{5 \cdot 6} + \frac{1}{6} = \frac{1}{20} + \frac{1}{30} + \frac{1}{6} = \frac{3}{60} + \frac{2}{60} + \frac{10}{60} = \frac{15}{60} = \frac{1}{4}</math>
: jadi lebih singkatnya adalah <math>\frac{\mathbf{1}}{\mathbf{4} \cdot 5} + \frac{1}{5 \cdot 6} + \frac{1}{6} = \frac{1}{4}</math>
=== Contoh 5A ===
Berapa nilai dari <math>1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4)</math>?
: <math> 1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) = 1 + 3 + 6 + 10 = 20</math>
: jadi lebih singkatnya adalah <math>1 \times 4 + 2 \times 3 + 3 \times 2 + 4 \times 1 = 2 \cdot (1 \times 4 + 2 \times 3) = 2 \cdot (4 + 6) = 20</math>
=== Contoh 5B ===
Berapa nilai dari <math>1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + (1 + 2 + 3 + 4 + 5)</math>?
: <math> 1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + (1 + 2 + 3 + 4 + 5) = 1 + 3 + 6 + 10 + 15 = 35</math>
: jadi lebih singkatnya adalah <math>1 \times 5 + 2 \times 4 + 3 \times 3 + 4 \times 2 + 5 \times 1 = 2 \cdot (1 \times 5 + 2 \times 4) + 3 \times 3 = 2 \cdot (5 + 8) + 9 = 35</math>
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