Is \(\displaystyle{1}-{\frac{{{1}}}{{{2}}}}+{\frac{{{1}}}{{{3}}}}-{\frac{{{1}}}{{{4}}}}+{\frac{{{1}}}{{{5}}}}-⋯={\ln{{2}}}\) true?

Kennedy Ford

Kennedy Ford

Answered question

2022-04-15

Is 112+1314+15=ln2 true?

Answer & Explanation

Gonarsu2dw8

Gonarsu2dw8

Beginner2022-04-16Added 19 answers

It is true and the Wikipedia article should have 1 at the end. Expansions like this have a radius of convergence, in this case 1. For x less than the radius the series converges absolutely. For x outside the series diverges. When x is equal to the radius it takes closer analysis. Here the series is conditionally convergent as we can show
i=1(-1)i+1i=i=112i-1-12i
i=112i(2i1)
which we know converges.

Kendall Wilkinson

Kendall Wilkinson

Beginner2022-04-17Added 17 answers

That the series expansion of ln(1+x) holds for x=1 is a consequence of Abel's Limit theorem

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