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

waigaK

waigaK

Answered question

2021-02-25

Consider the telescoping series
k=3(kk2)
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.

Answer & Explanation

Alannej

Alannej

Skilled2021-02-26Added 104 answers

Given the telescoping series
k=3(kk2)
a) The general term of the series is
ak=kk2
Take limit as k
limkak=limk(kk2)
Multiply the numerator and denominator with k+k+2. Then
limkak=limk(kk2)(k+k2)(k+k2)
=limk2(k+k2)
=0
The divergence test states if the nth term of a series nan does not converge to zero, then the series diverges.
Here ak0. 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=(332)+(442)+(552)
=31+42+53
=1+22+5
=12+5
s4=s3+(662)
=12+5+64
=12+5+62
=12+5+6
Also,
s5=s4+(772)
=12+5+6+75
=12+6+7
and
s6=s5+(882)

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