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Convergent Series

In a given sequence of numbers, the sum of the terms is called as the series. Let us consider a sequence $a_1, a_2, a_3, …$ where, the $S_n$ is called as the sum of first $n$ terms of the sequence terms.

$S_n = \sum_{k = 1}^n a_k$

Divergent series are those which do not converge. That is, it is not a convergent.

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Convergent and Divergent Series

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Suppose $\sum a_{n}$ is an infinite series. We define a sequence $<S_n>$ as follows:
$S_1 = a_1$
$S_2 = a_1 + a_2$
$S_3 = a_1 + a_2 + a_3$
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$S_n = a_1 + a_2 + ..........+ a_n$, and so on.
The sequence $<S_n>$ is called the sequence of partial sums of the series $\sum a_{n}$. The series $\sum a_{n}$ is said to be convergent if the sequence $<S_n>$ of partial sums is convergent and $\lim_{n\rightarrow \infty }S_{n}$ is called the sum of the series $\sum a_{n}$. The series $\sum a_{n}$ is said to be divergent if the sequence $<S_n>$ of partial sums is divergent. The series $\sum a_{n}$ is said to oscillate if the sequence $<S_n>$ of partial sums oscillates.

Convergent Infinite Series

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An infinite series $a_0 + a_1 + a_2 + a_3 + ..........$ is said to be converge if $\lim_{r\rightarrow \infty }a_{r} = 0$.
If we have
$S_0 = a_0$
$S_1 = a_1$
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$S_{n} = \sum_{r = 0}^{n}a_{r}$
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If the sequence $<S_n>$ converges to the given limit say $M$, then the given infinite series is said to converge to the partial sum $M$ and we can denote it as
$\sum_{r = 0}^{\infty}a_{r} = M$

Uniformly Convergent Series

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In mathematics, we can divide convergence in terms of point-wise convergent and uniformly convergent. If we have $f_{N} = \sum_{n =1}^{N}f_{n}(x)$ and $\lim_{N \rightarrow \infty }f_{N}(x) = f(x)$ $\forall$ x $\in$ A. Then, the sum is uniformly convergent in the given interval x $\in$ A if for every $\epsilon$ > 0 there exists any N such that n $\geq$ N we have $If(x) - f_n(x)I < \epsilon$. The same N work for all x $\in$ A.

There is a test for uniformly convergent which is known as Weierstress test. The Weierstress test states that if $f(x) = \sum_{n = 1}^{\infty}g_{n}(x)$ and if $Ig_n(x)I \leq L_n$ for all n $\geq$ N (any fixed integer). Then, $\forall$ x $\in$ A and if $\sum_{n = 1}^{\infty} L_{n}$ converges, then $\sum_{n = 1}^{\infty }g_{n}(x)$ converges uniformly to $f(x)$ over the interval x $\in$ A.

Convergent Power Series

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Any series $\sum \frac{1}{n^{p}} = \frac{1}{1^{p}} + \frac{1}{2^{p}} + \frac{1}{3^{p}} + .....$ where p > 0 converges if p > 1 and diverges if p $\leq$ 1. Power series or p-series is extensively used to examine the convergence and divergence of a large number of series.

Convergent Geometric Series

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A geometric series is a series which has constant ratio in between successive terms. $\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + .....$ is a geometric series since each successive term can be found by multiplying previous term by $\frac{1}{3}$.

Any series which can be expressed as $\sum_{n = 0}^{\infty}ar^{n} = a + ar + ar^{2} + ar^{3} +......$ is called geometric series. If $IrI < 1$, then the given series is said to converge to $\frac{a}{1 - r}$.

Convergent Series Examples

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A series is said to converge or convergent only when there is a limit $L$ such that for any positive number $\epsilon > 0$, there exists a large integer $N$ for all $n \in N$.

Examples of the divergent series involve the reciprocal of the positive integers.

(1 / 1) + (1 / 2 ) + (1 / 3) + (1 / 4) + ….-> L

If we alternate the signs of the positive integer’s reciprocal then we can get the convergent series.

(1 / 1) - (1 / 2 ) + (1 / 3) - (1 / 4) + (1 / 5) - (1 / 6) + …. = ln 2

The reciprocals of the factorials and the triangular numbers give the convergent series.

(1 / 1) + (1 / 1 ) + (1 / 2) + (1 / 6) + (1 / 24) + (1 / 120) +…= e

And the triangular numbers series include

(1 / 1) + (1 / 3 ) + (1 / 6) + (1 / 10) + (1 / 15) + (1 / 21) + ... = 2

The series
$\sum \frac{1}{n^{2}} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + .......$ converges.

The geometric series $1 + r + r^{2} + r^{3} + .......... (r > 0)$ is also convergent.

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