Converges or Diverges?

Calculus Level 2

Does the following series converge or diverge?

∑ c n n ! 2 , c ∈ R \sum \frac{c^n}{{n!}^2}, \quad c \in \mathbb R

Notation: ! ! denotes the factorial notation .

The series converges It can not be determined The series diverges

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Chew-Seong Cheong
Dec 19, 2019

∑ n = 0 ∞ c n n ! = e c \displaystyle \sum_{n=0}^\infty \frac {c^n}{n!} = e^c for all real c c . Therefore the series converges .

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Hana Wehbi
Dec 20, 2019

Define a n = c n ( n ! ) 2 a_n= \frac{c^n}{(n!)^2} then by the ratio test we get a n + 1 a n = c n + 1 ( ( n + 1 ) ! ) 2 × ( n ! ) 2 c n = c ( n + 1 ) 2 \frac{a_{n+1}}{a_n} = \frac{ c^{n+1}}{((n+1)!)^2} \times \frac{(n!)^2}{c^n} = \frac{c}{(n+1)^2}

c ( n + 1 ) 2 → 0 as n → ∞ \frac{c}{(n+1)^2} \rightarrow 0 \text{ as } n \rightarrow \infty thus the series converges.

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