Answer to Question #265711 in Calculus for Sayem

Question #265711

Determine whether the following series converge, converge absolutely, converge conditionally, or diverge.

  1. k=1\displaystyle\sum_{k=1}^ ∞ 3k/k2*2k
  2. k=1\displaystyle\sum_{k=1}^ ∞ (-1)k+1 / ( √(k+1) -√k)
  3. k=1\displaystyle\sum_{k=1}^ ∞ (-1)k+1 [√(k+1) -√k)]
1
Expert's answer
2021-11-22T17:40:25-0500

Let us determine whether the following series converge, converge absolutely, converge conditionally, or diverge.


1) k=13kk22k\displaystyle\sum_{k=1}^{\infty}\frac{3^k}{k^22^k} 


Let us use D'Alembert's Ratio Test.


Since limn(3k+1(k+1)22k+1:3kk22k)=limn3k22(k+1)2=limn3k22(k2+2k+1)=32>1,\lim\limits_{n\to\infty}(\frac{3^{k+1}}{(k+1)^22^{k+1}}:\frac{3^k}{k^22^k}) =\lim\limits_{n\to\infty}\frac{3k^2}{2(k+1)^2}=\lim\limits_{n\to\infty}\frac{3k^2}{2(k^2+2k+1)}=\frac{3}2>1,


we conclude that this series diverge.



2) Since

k=1(1)k+1k+1k=k=1(1)k+1(k+1+k)(k+1k)(k+1k)=k=1(1)k+1(k+1+k)((k+1)k)=k=1(1)k+1(k+1+k),\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{ \sqrt{k+1} -\sqrt{k}} =\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}(\sqrt{k+1} +\sqrt{k})}{ (\sqrt{k+1} -\sqrt{k})(\sqrt{k+1} -\sqrt{k})} =\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}(\sqrt{k+1} +\sqrt{k})}{ ((k+1)-k)} =\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}(\sqrt{k+1} +\sqrt{k}), 

and limk(k+1+k)=,\lim\limits_{k\to\infty}(\sqrt{k+1} +\sqrt{k})=\infty,

we conclude that the necessary condition of covergence is not met, and hence this series is divergent.



3) k=1(1)k+1(k+1k)\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}( \sqrt{k+1} -\sqrt{k})


Consider the series k=1(1)k+1(k+1k)=k=1(k+1k).\displaystyle\sum_{k=1}^{\infty}|(-1)^{k+1}( \sqrt{k+1} -\sqrt{k})|=\displaystyle\sum_{k=1}^{\infty}(\sqrt{k+1} -\sqrt{k}).

Since An=(21)+(32)+(43)++(n+1n)=n+11,A_n=(\sqrt{2} -1)+(\sqrt{3} -\sqrt{2})+(\sqrt{4} -\sqrt{3})+\ldots +(\sqrt{n+1} -\sqrt{n})=\sqrt{n+1} -1,

and limnAn=limn(n+11)=,\lim\limits_{n\to\infty}A_n=\lim\limits_{n\to\infty}(\sqrt{n+1} -1)=\infty, we conclude that the series k=1(1)k+1(k+1k)\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}( \sqrt{k+1} -\sqrt{k}) diverge absolutely.



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