infinite series - 5

[logistic_guy]

\(\displaystyle \textcolor{indigo}{\bold{Solve.}}\)

\(\displaystyle \sum_{k=0}^{\infty}(-1)^k\frac{3}{k!}\)

 

[Agent Smith]

3k, 0 < k < 1
Is it [imath]\pi[/imath]?

 

[logistic_guy]

\(\displaystyle \lim_{k\rightarrow \infty}\frac{3}{(k + 1)!}\frac{k!}{3} = \lim_{k\rightarrow \infty}\frac{k!}{(k + 1)k!} = \lim_{k\rightarrow \infty}\frac{1}{k + 1} = 0 < 1\)

Then,

\(\displaystyle \sum_{k=0}^{\infty}(-1)^k\frac{3}{k!}\) \(\displaystyle \ \textcolor{blue}{\text{converges.}}\)