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Investigate-the-series-1-1-2-1-2-3-1-3-4-1-4-5-Does-it-Converges-or-Diverges-




Question Number 184920 by Mastermind last updated on 13/Jan/23
Investigate the series  (1/(1×2))+(1/(2×3))+(1/(3×4))+(1/(4×5))+...  Does it Converges or Diverges?
$$\mathrm{Investigate}\:\mathrm{the}\:\mathrm{series} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}×\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{4}×\mathrm{5}}+… \\ $$$$\mathrm{Does}\:\mathrm{it}\:\mathrm{Converges}\:\mathrm{or}\:\mathrm{Diverges}? \\ $$
Answered by aba last updated on 14/Jan/23
S_n =Σ_(k=1) ^n (1/(k(k+1)))=Σ_(k=1) ^n ((1/k)−(1/(k+1)))=1−(1/(n+1))  Σ_(n=1) ^(+∞) =lim_(n→+∞) S_n =1  it converges
$$\mathrm{S}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}\left(\mathrm{k}+\mathrm{1}\right)}=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{k}}−\frac{\mathrm{1}}{\mathrm{k}+\mathrm{1}}\right)=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{+\infty} {\sum}}=\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}S}_{\mathrm{n}} =\mathrm{1} \\ $$$$\mathrm{it}\:\mathrm{converges} \\ $$

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