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Evaluate $\sum_{n=1}^\infty \left(e^{\frac{1}{n}} - 1\right)$

$ \lim e^{\frac{1}{n}} = 1$

$\sum_{n=1}^\infty \left(e^{\frac{1}{n}} - 1\right) > \sum_{n=1}^\infty(-1) = -\infty$

I'm not sure how to solve it. I tried two no convergence ways. what do u guys think about it?

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    $\begingroup$ is $-1$ inside or outside of the sum? $\endgroup$ – Alex Jan 3 '16 at 19:10
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Your idea of using a comparison is a good one, however, the comparison $$e^{\frac{1}n}-1>-1$$ is not particularly helpful, since the sum over $-1$ diverges to $-\infty$, so the comparison doesn't actually tell us anything. A more useful comparison would be $$e^{\frac{1}n}-1>\frac{1}n$$ since $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$ and if $x$ is positive, this is just $1+x+\text{positive terms}$ so $e^x>1+x$.

In general, notice that $f(x)=e^x-1$ is a function which has $f(0)=0$ and $f'(0)\neq 0$. One can use the bounds provided by the derivative to show that $\sum_{i=0}^{\infty}f(a_i)$ converges absolutely exactly when $\sum_{i=0}^{\infty}a_i$ converges absolutely.

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Since $e^x\geq 1+x$ we have $e^{1/n}-1\geq 1/n$ and hence the sum diverges by the comparison test.

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In THIS ANSWER, I showed using standard, non-calculus based tools that the exponential function satisfies the inequality

$$e^{x}\ge 1+x$$

Therefore, we have with $x=1/n$

$$e^{1/n}-1\ge \frac1n$$

Since the harmonic series, $\sum_{n=1}^\infty \frac1n$, diverges, then by the Comparison Test, the series of interest diverges.

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