Determine whether three series converges or diverges. if it converges, find its sum sum_(n=0)^(oo)((-1)^n)/4^n

Braxton Pugh 2021-02-21 Answered
Determine whether three series converges or diverges. if it converges, find its sum
n=0(1)n4n
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Expert Answer

brawnyN
Answered 2021-02-22 Author has 91 answers
1)Here the series is given by n=1(1)nan=n=0(1)n4n.This is a alternating series with an=14n.Carlyan is a decreasing sequence and
limnan=limn14n=0
Therefore by Leibnit'z rule the given series n=1(1)n4n is convergent.
2)Consider the series n=11n.The n-th partial sum of the series is given by
Sn=1+12+13++1n.
Now
S2n=1+12+13+14+15+16+17+18++12n
=1+12+(13+14)+(15+16+17+18)++12n
1+12+(14+14)+(18+18+18+18)++(1n++2n112n)
=1+12+12+12++12
=1+n2
but {n2} is a divergent sequence. This shows that S2n
is also divergent and hence the series n=11n is divergent.
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