Home » Blog » Find all z such that the series (-1)n / (z+n) converges

Find all z such that the series (-1)n / (z+n) converges

Find all complex numbers z such that the series

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{z+n} \]


First, the series is not absolutely convergent for any z \in \mathbb{Z} since

    \[ \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{z+n} \right| = \sum_{n=1}^{\infty} \frac{1}{|z+n|}. \]

By limit comparison with \sum \frac{1}{n} this always diverges.

Then, if z \neq -n for any positive integer n the series is convergent since \frac{1}{z+n} is monotonically decreasing and \lim_{n \to \infty} \frac{1}{z+n} = 0. If z = -n for some positive integer n, then the series is not defined since it is undefined for that term.


  1. Anonymous says:


    *** QuickLaTeX cannot compile formula:
    *** Error message:
    Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: unable to get local issuer certificate
    Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
    These links might help in finding solution:

    , and the real and imaginary parts of the original series are alternating series(es), which converge by Lebniz’s rule.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):