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Question Number 102745 by Dwaipayan Shikari last updated on 10/Jul/20
1−1+1−1+1−1+1−1+.....=(1/2)   {But it diverges  1+1+1+1+1+1+1+......=−(1/2)      {But it diverges  1+2+4+8+16+.....=−1                   {But it diverges  1+2+3+4+5+6+7=−(1/(12))                {But it diverges  1−2+4−8+.....=(1/3)                           {But it diverges  1−2+3−4+5−6+.....=(1/4)       {Is it a divergent?????
$$\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+\mathrm{1}−\mathrm{1}+…..=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left\{{But}\:{it}\:{diverges}\right. \\ $$$$\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+\mathrm{1}+……=−\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\left\{{But}\:{it}\:{diverges}\right. \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{4}+\mathrm{8}+\mathrm{16}+…..=−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{{But}\:{it}\:{diverges}\right. \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}=−\frac{\mathrm{1}}{\mathrm{12}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{{But}\:{it}\:{diverges}\right. \\ $$$$\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+…..=\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left\{{But}\:{it}\:{diverges}\right. \\ $$$$\mathrm{1}−\mathrm{2}+\mathrm{3}−\mathrm{4}+\mathrm{5}−\mathrm{6}+…..=\frac{\mathrm{1}}{\mathrm{4}}\:\:\:\:\:\:\:\left\{{Is}\:{it}\:{a}\:{divergent}?????\right. \\ $$
Commented by Dwaipayan Shikari last updated on 10/Jul/20
Funny great sequences
$${Funny}\:{great}\:{sequences} \\ $$
Commented by prakash jain last updated on 10/Jul/20
See the link below to get an idea behind theory for calculating these sum of divergent series. https://en.m.wikipedia.org/wiki/Analytic_continuation also this link given some additional info about other summation methods https://en.m.wikipedia.org/wiki/Divergent_series

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