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I am trying to determine the value of the following infinite series:

$$\sum_{n=0}^\infty \left((-1)^n * \left(\frac{n}{2}+\frac{5}{4}\right) + \frac{3}{4}\right)$$

This series is equal to the infinite series $2 - 1 + 3 - 2 +....$

I believe this series diverges to positive infinity since the difference between any two consecutive terms is 1, so $1 + 1 + 1 + ... = \infty$. However, if we split the formula to positive and negative terms and consider these groups of terms individiually, we get $\infty - \infty = 0$ since both series are divergent. Since both series are divergent, any finite number of terms enumerated from the positive series can be subtracted away by the negative series, even if it takes longer for the negative series to catch up than the reverse scenario.

Now, I know that one of these two reasonings is correct and the other is incorrect. Which one is correct and why?

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  • $\begingroup$ $$\infty-\infty$$ is not zero $\endgroup$ – Dr. Sonnhard Graubner Jun 5 '17 at 17:26
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    $\begingroup$ Be careful. The difference of two divergent series is not 0. Rearranging terms is only valid when the series converges absolutely. $\endgroup$ – NDewolf Jun 5 '17 at 17:26
  • $\begingroup$ The series is not convergent, by the term test. In short: don't go around grouping terms together as you please -- this can lead to pretty much anything. $\endgroup$ – Clement C. Jun 5 '17 at 17:27
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    $\begingroup$ Note: if $\infty - \infty = 0$ and $1 + \infty = \infty$, then $$ 0 = (1 + \infty) - \infty = 1 + (\infty - \infty) = 1 $$ $\endgroup$ – Omnomnomnom Jun 5 '17 at 17:33
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    $\begingroup$ It diverges to $+\infty$. $\endgroup$ – NDewolf Jun 5 '17 at 17:43
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The second term between brackets will always cancel out between two consecutive steps in the series. However the first term between and brackets and the last term in the series do diverge when you sum from 0 to $+\infty$.

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