For each series, find an explicit formula for the sequence of partial sums and determine if the series converges. sum_{n=1}^inftyfrac{1}{n(n+1)}

illusiia 2021-02-18 Answered
For each series, find an explicit formula for the sequence of partial sums and determine if the series converges.
n=11n(n+1)
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Expert Answer

joshyoung05M
Answered 2021-02-19 Author has 97 answers
Given: The series n=11n(n+1)
To determine: The explicit formula for the sequence of partial sum and convergence of the series.
Explanation:
The general term (k-th term) of the sequence of partial sum is of a series is the sum of first k terms.
Here the given series is n=11n(n+1)
So general term of sequence of partial sum of this series is n=1n=k1n(n+1)
This general can be re-written as
tk=n=1n=k1n(n+1)
tk=n=1n=k(1n1n+1)
tk=(1112)+(1213)+(1314)+...+(1k1k+1)
tk=(11k+1)
Hence general term of sequence of partial sum of this series is tk=(11k+1)
So sequence of partial sum is {11n+1}n=1
Now it is known that n=1an=limkn=1kan
Therefore here
n=11n(n+1)=limkn=1k1n(n+1)
n=11n(n+1)=limk(11k+1)
n=11n(n+1)=1
Hence the series is convergent as converges to 1.
Answer:
The sequence of partial sum is {11n+1}n=1
And the series is convergent as converges to 1.
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Jeffrey Jordon
Answered 2021-12-16 Author has 2070 answers

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