Question

How do you know when to use the integral test for an infinite series?

Answer 1

The integral test for convergence is a method used to test the infinite series of non-negative terms for convergence. It is also known as Maclaurin-Cauchy Test. The integral test for an infinite series is based on the conditions of the terms, whether it converges or divergences.

Complete step by step solution:
To find when to use the integral test for an infinite series some points to be noted are as follows:
The integral comparison test is mainly for the integral terms. If we have two functions say f(x) and g(x) in such a way that $g\left( x \right) \geqslant f\left( x \right)$ on the given interval $\left[ {c,\infty } \right]$, then it should have the following conditions.
If the term $\int\limits_c^\infty {g\left( x \right)} dx$ converges, then the term so does $\int\limits_c^\infty {f\left( x \right)} dx$.
If the term $\int\limits_c^\infty {f\left( x \right)} dx$ divergences, then the term so does $\int\limits_c^\infty {g\left( x \right)} dx$.

Additional information:
Integral Test for Convergence
The integral test for convergence is a method used to test the infinite series of non-negative terms for convergence. It is also known as Maclaurin-Cauchy Test.
Let N be a natural number (non-negative number), and it is a monotonically decreasing function, then the function is defined as
$f:\left[ {N,\infty } \right] \to R$
Then the series $\sum\limits_{m = N}^\infty {f\left( m \right)}$ is convergent if and only if the integral $\int\limits_N^\infty {f\left( t \right) \cdot } dt$ is finite.
If you can define f so that it is a continuous, positive, decreasing function from 1 to infinity (including 1) such that $a\left[ n \right] = f\left( n \right)$, then the sum will converge if and only if the integral of function f from 1 to infinity converges.

Note: The key point to find the integral test is these integrals are convergent if the associated limit exists and is a finite number and divergent if the associated limit either doesn’t exist or is (plus or minus) infinity. The integral test is used to find whether the given series is convergence or not. The convergence of series is more significant in many situations when the integral function has the sum of a series of functions.