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.