# For which positive integers k is the following series convergent?

$$\sum\limits_{n=1}^{\infty}\dfrac{(n!)^2}{(kn)!}$$

This question aims to find the value of the positive integer $k$, for which the given series is convergent.

A series in mathematics is a representation of the procedure of adding infinite quantities sequentially to a given starting quantity. The series analysis is an important part of calculus and its generalization such as mathematical analysis. A convergent series is one in which the partial sums approach a particular number usually known as a limit. A divergent series is one in which the partial sums do not tend to a limit. Divergent series usually tend to positive or negative infinity and do not tend to a particular number.

The Ratio Test aids in determining whether a series converges or diverges. Consider the series $\sum a_n$. The Ratio Test examines $\lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$ to determine the long term behavior of the series. As $n$ approaches infinity, this ratio compares the value of $a_{n+1}$ to the previous term $a_n$ to determine the amount of decrease in terms. If this limit is more than one, then $\left|\dfrac{a_{n+1}}{a_n}\right|$ will show that the series is not decreasing for all values of $n$ after a particular point. In this case, the series is said to be divergent. However, if this limit is smaller than one, absolute convergence can be observed in the series.

## Expert Answer

Since the series is convergent, so by the Ratio Test:

$\left|\dfrac{a_{n+1}}{a_n}\right|=\dfrac{\dfrac{[(n+1)!]^2}{[k(n+1)]!}}{\dfrac{(n!)^2}{(kn)!}}$

$=\dfrac{[(n+1)!]^2}{[k(n+1)]!}\times \dfrac{(kn)!}{(n!)^2}$

$=\dfrac{[(n+1)\cdot n!]^2}{(kn+k)!}\times \dfrac{(kn)!}{(n!)^2}$

$=\dfrac{(n+1)^2\cdot (n!)^2}{(kn+k)\cdots (kn+2)\cdot (kn+1)(kn)!}\times \dfrac{(kn)!}{(n!)^2}$

$=\dfrac{(n+1)^2}{(kn+k)\cdots (kn+2)\cdot (kn+1)}$

Now, for $k=1$:

$\left|\dfrac{a_{n+1}}{a_n}\right|=\dfrac{(n+1)^2}{n+1}=n+1$

And so,  $\lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|=\lim\limits_{n\to\infty}(n+1)=\infty$

Hence, the series diverges for $k=1$.

For $k=2$ we have:

$\left|\dfrac{a_{n+1}}{a_n}\right|=\dfrac{(n+1)^2}{(2n+1)(2n+2)}=\dfrac{n^2+2n+1}{4n^2+6n+2}$

And,  $\lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|=\lim\limits_{n\to\infty}\dfrac{n^2+2n+1}{4n^2+6n+2}=\dfrac{1}{4}<1$

Hence, the series converges for $k=2$. We will have a function where the degree of the numerator will be smaller than the degree of the denominator for $k>2$. So, the limit becomes $0$ for $n$ approaching to $\infty$. Finally, it can be concluded that the given series converges for all $k\geq 2$.

## Example 1

Determine whether the series  $\sum\limits_{n=1}^{\infty}\dfrac{(-15)^n}{3^{n+2}n}$ converges or diverges.

### Solution

Let $a_n=\dfrac{(-15)^n}{3^{n+2}n}$

So, $a_{n+1}=\dfrac{(-15)^{n+1}}{3^{n+3}(n+1)}$

Suppose that  $L=\lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$

$L=\lim\limits_{n\to\infty}\left|\dfrac{(-15)^{n+1}}{3^{n+3}(n+1)}\cdot \dfrac{3^{n+2}n}{(-15)^n}\right|$

$L=\lim\limits_{n\to\infty}\left|\dfrac{-15n}{3(n+1)}\right|$

$L=\dfrac{15}{3}\lim\limits_{n\to\infty}\dfrac{n}{(n+1)}$

$L=\dfrac{15}{3}\lim\limits_{n\to\infty}\dfrac{n}{n(1+\frac{1}{n})}$

$L=\dfrac{15}{3}\lim\limits_{n\to\infty}\dfrac{1}{(1+\frac{1}{n})}$

$L=\dfrac{15}{3}\dfrac{1}{(1+\frac{1}{\infty})}$

$L=\dfrac{15}{3}\dfrac{1}{(1+0)}$

$L=\dfrac{15}{3}(1)$

$L=\dfrac{15}{3}$

$L=5>1$

So by Ratio Test, the given series is divergent.

## Example 2

Test the series  $\sum\limits_{n=1}^{\infty}\dfrac{n!}{2^n}$, for convergence or divergence.

### Solution

Let $a_n=\dfrac{n!}{2^n}$

So, $a_{n+1}=\dfrac{(n+1)!}{2^{n+1}}$

Let  $L=\lim\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$

$L=\lim\limits_{n\to\infty}\left|\dfrac{(n+1)!}{2^{n+1}}\cdot \dfrac{2^n}{n!}\right|$

$L=\lim\limits_{n\to\infty}\left|\dfrac{(n+1)n!}{2^n\cdot 2^1}\cdot \dfrac{2^n}{n!}\right|$

$L=\lim\limits_{n\to\infty}\dfrac{n+1}{2}$

$L=\infty>1$

Since the limit is equal to infinity, therefore, the given series is divergent by Ratio Test.

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