convergence test for infinite series

Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series 

If the limit of the summand is undefined or nonzero, that is ${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{�}_{�}\ne 0}$, then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.

Ratio test

This is also known as d'Alembert's criterion.

Suppose that there exists ${\displaystyle �}$ such that

${\displaystyle \underset{�\to \mathrm{\infty }}{lim}\left|\frac{{�}_{�+1}}{{�}_{�}}\right|=�.}$

If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Root test

This is also known as the nth root test or Cauchy's criterion.

Let

${\displaystyle �=\underset{�\to \mathrm{\infty }}{lim sup}\sqrt[�]{|{�}_{�}|},}$

where ${\displaystyle lim sup}$ denotes the limit superior (possibly ${\displaystyle \mathrm{\infty }}$; if the limit exists it is the same value).

If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.^[1]

Integral test

The series can be compared to an integral to establish convergence or divergence. Let ${\displaystyle �:[1,\mathrm{\infty })\to {\mathbb{�}}_{+}}$ be a non-negative and monotonically decreasing function such that ${\displaystyle �(�)={�}_{�}}$. If

then the series converges. But if the integral diverges, then the series does so as well. In other words, the series ${\displaystyle {�}_{�}}$ converges if and only if the integral converges.

p-series test

A commonly-used corollary of the integral test is the p-series test. Let . Then ${\displaystyle \sum _{�=�}^{\mathrm{\infty }}(\frac{1}{{�}^{�}})}$ converges if .

The case of ${\displaystyle �=1,�=1}$ yields the harmonic series, which diverges. The case of ${\displaystyle �=2,�=1}$ is the Basel problem and the series converges to ${\displaystyle \frac{{�}^2}{6}}$. In general, for , the series is equal to the Riemann zeta function applied to ${\displaystyle �}$, that is ${\displaystyle �(�)}$.

Direct comparison test

If the series ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}{�}_{�}}$ is an absolutely convergent series and ${\displaystyle |{�}_{�}|\le |{�}_{�}|}$ for sufficiently large n , then the series ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}{�}_{�}}$ converges absolutely.

Limit comparison test

If , (that is, each element of the two sequences is positive) and the limit ${\displaystyle \underset{�\to \mathrm{\infty }}{lim}\frac{{�}_{�}}{{�}_{�}}}$ exists, is finite and non-zero, then ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}{�}_{�}}$ diverges if and only if ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}{�}_{�}}$ diverges

Alternating series test

Suppose the following statements are true:

• ${\displaystyle {�}_{�}}$ are all positive,
• ${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{�}_{�}=0}$ and
• for every n, ${\displaystyle {�}_{�+1}\le {�}_{�}}$.

Then ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}(-1{)}^{�}{�}_{�}}$ and ${\displaystyle \sum _{�=1}^{\mathrm{\infty }}(-1{)}^{�+1}{�}_{�}}$ are convergent series. This test is also known as the Leibniz criterion.

Power series

In mathematics, a power series (in one variable) is an infinite series of the form

$${\displaystyle \sum _{�=0}^{\mathrm{\infty }}{�}_{�}{\left(�-�\right)}^{�}={�}_0+{�}_1(�-�)+{�}_2(�-�{)}^2+\dots }$$

where a_n represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.

$${\displaystyle \sum _{�=0}^{\mathrm{\infty }}{�}_{�}{�}^{�}={�}_0+{�}_1�+{�}_2{�}^2+\dots }$$

In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

Radius of convergence

$\sum _{�=0}^{\mathrm{\infty }}{�}_{�}(�-�{)}^{�}$ is convergent for some values of the variable x

if

 |x – c| < r     ,  converges

 |x – c| > r     ,  diverges  

 r is called the radius of convergence of the power series; in general it is given as

or 

For better understanding you can refer this video