Original & Alternating\text{Original \& Alternating}

1+12+13+14+15+16+17+18+=112+1314+1516+1718+=ln21+13+15+17+19+111+113+115+=113+1517+19111+113115+=π41+12+14+18+116+132+164+1128+=2112+1418+116132+1641128+=231+122+132+142+152+162+172+182+=π261122+132142+152162+172182+=π2121+12!+13!+14!+15!+16!+17!+18!+=e1112!+13!14!+15!16!+17!18!+=11e1+13!+15!+17!+19!+111!+113!+115!+=sinh1113!+15!17!+19!111!+113!115!+=sin11+12!+14!+16!+18!+110!+112!+114!+=cosh1112!+14!16!+18!110!+112!114!+=cos11+2+3+4+5+6+7+8+=11212+34+56+78+=14\begin{aligned} 1 + \dfrac12 + \dfrac13 + \dfrac14 + \dfrac15 + \dfrac16 + \dfrac17 + \dfrac18 + \dots &= \infty \\ 1 - \dfrac12 + \dfrac13 - \dfrac14 + \dfrac15 - \dfrac16 + \dfrac17 - \dfrac18 + \dots &= \ln2 \\ \\ 1 + \dfrac13 + \dfrac15 + \dfrac17 + \dfrac19 + \dfrac{1}{11} + \dfrac1{13} + \dfrac1{15} + \dots &= \infty \\ 1 - \dfrac13 + \dfrac15 - \dfrac17 + \dfrac19 - \dfrac{1}{11} + \dfrac1{13} - \dfrac1{15} + \dots &= \dfrac{\pi}{4} \\ \\ 1 + \dfrac12 + \dfrac14 + \dfrac18 + \dfrac1{16} + \dfrac{1}{32} + \dfrac1{64} + \dfrac1{128} + \dots &= 2 \\ 1 - \dfrac12 + \dfrac14 - \dfrac18 + \dfrac1{16} - \dfrac{1}{32} + \dfrac1{64} - \dfrac1{128} + \dots &= \dfrac23 \\ \\ 1 + \frac1{2^2} + \dfrac1{3^2} + \dfrac1{4^2} + \dfrac1{5^2} + \dfrac1{6^2} + \dfrac1{7^2} + \dfrac1{8^2} + \dots &= \dfrac{\pi^2}{6} \\ 1 - \frac1{2^2} + \dfrac1{3^2} - \dfrac1{4^2} + \dfrac1{5^2} - \dfrac1{6^2} + \dfrac1{7^2} - \dfrac1{8^2} + \dots &= \dfrac{\pi^2}{12} \\ \\ 1 + \frac1{2!} + \dfrac1{3!} + \dfrac1{4!} + \dfrac1{5!} + \dfrac1{6!} + \dfrac1{7!} + \dfrac1{8!} + \dots &= e - 1 \\ 1 - \frac1{2!} + \dfrac1{3!} - \dfrac1{4!} + \dfrac1{5!} - \dfrac1{6!} + \dfrac1{7!} - \dfrac1{8!} + \dots &= 1 - \dfrac1e \\ \\ 1 + \frac1{3!} + \dfrac1{5!} + \dfrac1{7!} + \dfrac1{9!} + \dfrac1{11!} + \dfrac1{13!} + \dfrac1{15!} + \dots &= \sinh1 \\ 1 - \frac1{3!} + \dfrac1{5!} - \dfrac1{7!} + \dfrac1{9!} - \dfrac1{11!} + \dfrac1{13!} - \dfrac1{15!} + \dots &= \sin1 \\ \\ 1 + \frac1{2!} + \dfrac1{4!} + \dfrac1{6!} + \dfrac1{8!} + \dfrac1{10!} + \dfrac1{12!} + \dfrac1{14!} + \dots &= \cosh1 \\ 1 - \frac1{2!} + \dfrac1{4!} - \dfrac1{6!} + \dfrac1{8!} - \dfrac1{10!} + \dfrac1{12!} - \dfrac1{14!} + \dots &= \cos1 \\ \\ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \dots &= \dfrac{-1}{12} \\ 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \dots &= \dfrac{1}{4} \\ \end{aligned}

Source: blackpenredpen\text{Source: blackpen}\textcolor{#D61F06}{redpen}

Note by Adhiraj Dutta
2 weeks, 5 days ago

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Really cool! @Adhiraj Dutta

Yajat Shamji - 2 weeks, 4 days ago

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Glad you liked it @Yajat Shamji

Adhiraj Dutta - 2 weeks, 4 days ago

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If you sourced BlackpenRedPen, he mentions that the second-last series has value 18\dfrac{-1}{8} and not 112\dfrac{-1}{12}, although there can be many more values according to algebraic manipulations. Afterall =+1\infty= \infty+ 1

Mahdi Raza - 1 week, 6 days ago

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I used Ramanujan's summation.

Adhiraj Dutta - 1 week, 5 days ago

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