Convergence of the series 
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n-1}{\frac {2}{2k+1}}\left({\frac {x-1}{x+1}}\right)^{2k+1}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><munderover><mo>∑<!-- ∑ --></mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−<!-- − --></mo><mn>1</mn></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><msup><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>x</mi><mo>−<!-- − --></mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n-1}{\frac {2}{2k+1}}\left({\frac {x-1}{x+1}}\right)^{2k+1}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1031c97e1fb5b23369848c4044a8c8d9a91e7b96" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:24.556ex; height:7.509ex;" alt="{\displaystyle \sum _{k=0}^{n-1}{\frac {2}{2k+1}}\left({\frac {x-1}{x+1}}\right)^{2k+1}}"></span></dd></dl>
towards <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ln</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span> with an increasing number of terms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>n</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle n}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>.
Marcel Marnitz, completely reworked by <a href="//commons.wikimedia.org/wiki/User:Georg-Johann" title="User:Georg-Johann">Georg-Johann</a>
2009-02-07
<span class="int-own-work" lang="en">Own work</span>