Solved: "Prove: the complex series ∑n=0∞(zn+1n+1−2z2n+32n+3) is discontinuous at..."

beuz89100g6c

Answered question

2022-02-28

Prove: the complex series

\sum_{n = 0}^{\infty} (\frac{z^{n + 1}}{n + 1} - \frac{2 z^{2 n + 3}}{2 n + 3})

is discontinuous at

z = 1

Mikayla Swan

Beginner2022-03-01Added 9 answers

For

| z | < 1

\sum_{\frac{z^{n + 1}}{n + 1}} - \frac{2 z^{2 n + 3}}{2 n + 3} = \sum \int t^{n} - 2 t^{2 n + 2}

= \int \sum t^{n} - 2 t^{2 n + 2} = \int \frac{1}{1 - t} - \frac{2 t^{2}}{1 - t^{2}}

= \int 2 - \frac{1}{1 + t} = 2 z - \ln (1 + z) = 2 - \ln 2

For

z = 1

\sum {\frac{1}{n + 1}} - \frac{2}{2 n + 3} = 2 \sum \frac{1}{2 n + 2} - \frac{1}{2 n + 3}

= 2 (1 - \sum \frac{{(- 1)}^{n}}{n + 1}) = 2 (1 - \ln 2) = 2 - 2 \ln 2 \neq 2 - \ln 2

Hence the discontinuity.

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