Consider the telescoping series ∑k=3∞(k−k−2) a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series? b) Write out the partial sums s3,s4,s5 and s6. c) Compute the n-th partial sum sn, and put it in closed form.

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Consider the telescoping series  ∑k=3∞(k−k−2)  a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series?  b) Write out the partial sums s3,s4,s5 and s6.  c) Compute the n-th partial sum sn, and put it in closed form.

Alannej · Accepted Answer

Given the telescoping series  ∑k=3∞(k−k−2)  a) The general term of the series is  ak=k−k−2  Take limit as k→∞  limk→∞ak=limk→∞(k−k−2)  Multiply the numerator and denominator with k+k+2. Then  limk→∞ak=limk→∞(k−k−2)(k+k−2)(k+k−2)  =limk→∞2(k+k−2)  =0  The divergence test states if the nth term of a series ∑nan does not converge to zero, then the series diverges.  Here ak→0. So, the series may or may not converge.  b) The n-th partial sum of series ∑nan is sum of the first n-terms of the series.  So,  s3=(3−3−2)+(4−4−2)+(5−5−2)  =3−1+4−2+5−3  =−1+2−2+5  =1−2+5  s4=s3+(6−6−2)  =1−2+5+6−4  =1−2+5+6−2  =−1−2+5+6  Also,  s5=s4+(7−7−2)  =−1−2+5+6+7−5  =−1−2+6+7  and  s6=s5+(8−8−2)

Consider the telescoping series sum_{k=3}^infty(sqrt{k}-sqrt{k-2}) a) Apply the Divvergence Test to the series. Show all details. What conclusions, if

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