# Tagged Questions

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### Showing how this infinite sum diverges

$\sum_{n=1}^{\infty} ((1+\frac{1}{n})^n - e)$ I tried both root and ratio tests (for the root test, the expression became way too complex to handle) but both didn't really work out. Would there be ...
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### If $\sum a_n$ converges and for almost all n $a_n>0$, does it mean that the series $\sum a_n$ converges absolutely?

If $\sum a_n$ converges and for almost all n $a_n>0$, does it mean that the series $\sum a_n$ converges absolutely ? Thanks.
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### Proving inequality, then use comparison test for convergence

My problem is formulated like this: Given that both $\sum_{n=1}^{\infty}a_{n}^2$ and $\sum_{n=1}^{\infty}b_{n}^2$ converges, establish the convergence (implies that it converges?) of the following ...
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### Limit comparison Test with a suitable $b_k$

My question is as follows. Does there exist $b_k$ such that $\displaystyle\lim_{k \to \infty}\frac{\left(\frac{\sin^2k}{k^2}\right)}{b_k}$ is a finite positive number and $\sum_{k=1}^{\infty}b_k$ ...
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### If $\sum a_n$ converges, so does $\sum a_n^{\frac{n}{n+1}}$

Let $a_n$ be a positive real sequence such that the series $\sum a_n$ converges. I was asked to prove that under such circumstances $\sum a_n^{\frac{n}{n+1}}$ converges. The previous sum can be ...
### Limit of $a_0 = 3 ; a_n = a_{n-1} + \frac{n-1}{n^2}$
I need to find the limit of $a_n$ for $n \rightarrow \infty$ but I am not sure how I would do it. $$a_0 = 3 ; a_n = a_{n-1} + \frac{n-1}{n^2}$$ I tried to transform $a_n$ to a non recursive ...
If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? \$\sum_{n=0}^{\infty} ...