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#2: Post edited
by
Snoopy
·
2024-08-06T21:28:30Z (5 months ago)
Find the value of $\displaystyle \sum_{k=1}^\infty\frac{k^2}{k!}$
- Find the value of $\sum_{k=1}^\infty\frac{k^2}{k!}$
#1: Initial revision
by
Snoopy
·
2024-08-06T21:28:16Z (5 months ago)
Find the value of $\displaystyle \sum_{k=1}^\infty\frac{k^2}{k!}$
**Problem.** Find the value of $\displaystyle \sum_{k=1}^\infty\frac{k^2}{k!}$.
**Note**: For most convergent series, proving convergence significantly differs from calculating their values, if it is even possible. One has tools such as [convergence tests](https://en.wikipedia.org/wiki/Convergence_tests) to determine whether a given series converges. On the other hand, finding the value of a convergent series is often highly nontrivial, as it frequently requires computing the value in a specific context. For calculus beginners, computing the values of convergent series can be particularly challenging, as they often struggle to place the problem within a broader context.
I will write my answers below. Answers from different perspectives are welcome.