Find the positive values of p for which the series converges.

Question

Find the positive values of p for which the series  converges. ​

Quizplus · Accepted Answer

The given series can be written in the form of general term as 
$a_n= \frac{p^n}{n!}$
Now, using the ratio test, 
$\frac{a_{n+1}}{a_n}=\frac{p}{n+1}$
For the series to converge, the above ratio should be less than 1. Therefore, 
$\frac{p}{n+1}<1$
Solving the above inequality, we get
$p<n+1$
As the series starts from n=0, we can replace n with n+1 in the above inequality to get 
$p<(n+1)+1=n+2$
Therefore, the series converges for values of p less than n+2 for any n. Since p has to be positive, we can take the smallest value of n i.e. n=0. So, p<2. 
Hence, the series converges only for values of p less than 2. 
Out of the given answer options, only option D satisfies this condition.

Find the Positive Values of P for Which the Series