Graph showing how the <a href="https://en.wikipedia.org/wiki/Mercator_Series" class="extiw" title="w:Mercator Series">Mercator Series</a> converges to <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 \left(x\right)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ln</mi><mo>⁡<!-- ⁡ --></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \ln \left(x\right)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436279c598d65b0b6212e3854d7cf726d603971e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln \left(x\right)}"></span> for <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 0&lt;x\leq 2}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mn>0</mn><mo>&lt;</mo><mi>x</mi><mo>≤<!-- ≤ --></mo><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle 0&lt;x\leq 2}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5e774e8f684c113ce583774c026c742972be05" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:9.852ex; height:2.343ex;" alt="{\displaystyle 0&lt;x\leq 2}"></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> included in the finite <a href="https://en.wikipedia.org/wiki/series_(mathematics)" class="extiw" title="w:series (mathematics)">series</a>
<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=1}^{n}{\frac {1}{k}}(x-1)^{k}\,.}"><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>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>k</mi></mfrac></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−<!-- − --></mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}(x-1)^{k}\,.}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826b5eaaae17acae452cdb052674f1d40bd09a5f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:15.054ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}(x-1)^{k}\,.}"></span></dd></dl>
<a href="//commons.wikimedia.org/wiki/User:Slaunger" title="User:Slaunger">Kim Hansen</a>
CC BY-SA 3.0
Screen shot of own work using the GFDL program <a rel="nofollow" class="external text" href="http://www.padowan.dk/graph/">Graph</a>.