Question

When testing for convergence, how do you determine which test to use?

Answer 1

Hint : When we get further and further in a sequence, the terms get closer and closer to a specific limit, that is, when adding the terms one after the other, if we get partial sums that become closer and closer to a given number, then the series converges and it is known as the convergence of the series.

Complete step-by-step answer :
There are various tests for determining the convergence of a series, they are as follows –
I.Divergence test –
If $\mathop {\lim }\limits_{n \to \infty } {a_n} \ne 0$ then $\sum\limits_n {{a_n}}$ diverges.

II.Integral test –
If ${a_n} = f(n)$ , $f(x)$ is a non-negative non-increasing function, then the condition for $\sum\limits_n^\infty {{a_n}}$ to converge is that the integral $\int\limits_1^\infty {f(x)} dx$ converges.

III.Comparison test –
If a series is similar to another p-series or geometric series then this test is used. The series should be a positive-term series.
If ${a_n} \leqslant {b_n}$ and $\sum {{b_n}}$ converges then $\sum {{a_n}}$ also converges.
If ${b_n} \leqslant {a_n}$ and $\sum {{b_n}}$ diverges then $\sum {{a_n}}$ also diverges.

IV.Limit comparison test –
If $\sum {{a_n}}$ and $\sum {{b_n}}$ are both positive-term series and $\mathop {\lim }\limits_{n \to \infty } \dfrac{{{a_n}}}{{{b_n}}} = L$ , where $0 < L < \infty$ then either $\sum {{a_n}}$ and $\sum {{b_n}}$ both converge or both diverge.

V.Alternating series test –
When we have an alternating series, that is, the series has alternative signs then we can write $\sum\limits_n^\infty {{a_n}} = \sum\limits_n^\infty {{{( - 1)}^n}{b_n}}$
If ${b_n} > 0$ , ${b_{n + 1}} \leqslant {b_n}$ and $\mathop {\lim }\limits_{n \to \infty } {b_n} = 0$ , then $\sum {{{( - 1)}^{n + 1}}{b_n}}$ converges.

VI.Ratio test –
In this test for a series $\sum {{a_n}}$ , $L = \mathop {\lim }\limits_{n \to \infty } \dfrac{{\left| {{a_{n + 1}}} \right|}}{{\left| {{a_n}} \right|}}$
If $L < 1$ then $\sum {{a_n}}$ converges.
If $L = 1$ then L doesn’t exist and thus test fails.
If $L > 1$ then $\sum {{a_n}}$ diverges.

VII.Root test –
For a series $\sum {{a_n}}$
Let $L = \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left| {{a_n}} \right|}}$
If $L < 1$ then $\sum {{a_n}}$ converges.
If $L = 1$ then L doesn’t exist and thus test fails.
If $L > 1$ then $\sum {{a_n}}$ diverges.
Hence we can use any of these tests depending on the series, however we use the ratio test the most.

Note : A series is defined as an expression in which infinitely many terms are added one after the other to a given starting quantity. It is represented as $\sum\limits_{n = 1}^\infty {{a_n}}$ where $\sum {}$ sign denotes the summation sign which indicates the addition of all the terms. We can also find the radius of convergence after testing whether the series converges or not.