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Question
\sum _{n=2}^{+\infty}\frac{n^{2}}{\left(2n-1\right)!}
Determine the convergence or divergence
\text{Converges}
Evaluate
\sum _{n=2}^{+\infty}\frac{n^{2}}{\left(2n-1\right)!}
Find the limit
\lim _{n\rightarrow +\infty}\left(\left|\frac{\frac{\left(n+1\right)^{2}}{\left(2n+1\right)!}}{\frac{n^{2}}{\left(2n-1\right)!}}\right|\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\left|\frac{\left(n+1\right)^{2}}{2n^{3}\left(2n+1\right)}\right|\right)
Remove the absolute value bars
\lim _{n\rightarrow +\infty}\left(\frac{\left(n+1\right)^{2}}{2n^{3}\left(2n+1\right)}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(\frac{\left(n\left(1+\frac{1}{n}\right)\right)^{2}}{n^{4}\left(4+\frac{2}{n}\right)}\right)
Calculate
\lim _{n\rightarrow +\infty}\left(\frac{n^{2}\left(1+\frac{1}{n}\right)^{2}}{n^{4}\left(4+\frac{2}{n}\right)}\right)
Reduce the fraction
\lim _{n\rightarrow +\infty}\left(\frac{\left(1+\frac{1}{n}\right)^{2}}{n^{2}\left(4+\frac{2}{n}\right)}\right)
Rewrite the expression
\frac{\lim _{n\rightarrow +\infty}\left(\left(1+\frac{1}{n}\right)^{2}\right)}{\lim _{n\rightarrow +\infty}\left(n^{2}\left(4+\frac{2}{n}\right)\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\left(1+\frac{1}{n}\right)^{2}\right)
\text{Use }\lim_{x\rightarrow c}(f(x)^a) = (\lim_{x\rightarrow c} f(x))^a\text{ to transform the expression}
\left(\lim _{n\rightarrow +\infty}\left(1+\frac{1}{n}\right)\right)^{2}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(1+\frac{1}{n}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(1\right)+\lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Calculate
1+\lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Calculate
1+0
Calculate
1
1^{2}
Evaluate the power
1
\frac{1}{\lim _{n\rightarrow +\infty}\left(n^{2}\left(4+\frac{2}{n}\right)\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(n^{2}\left(4+\frac{2}{n}\right)\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(n^{2}\right)\times \lim _{n\rightarrow +\infty}\left(4+\frac{2}{n}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(n^{2}\right)
Rewrite the expression
\left(\lim _{n\rightarrow +\infty}\left(n\right)\right)^{2}
Calculate
\left(+\infty\right)^{2}
Calculate
+\infty
\left(+\infty\right)\times \lim _{n\rightarrow +\infty}\left(4+\frac{2}{n}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(4+\frac{2}{n}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(4\right)+\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Calculate
4+\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Calculate
4+0
Calculate
4
\left(+\infty\right)\times 4
Simplify
+\infty
\frac{1}{+\infty}
Calculate
0
Solution
\text{Converges}
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