For each series, find an explicit formula for the sequence of partial sums and determine if the series converges. ∑n=1∞1n(n+1)

Question

For each series, find an explicit formula for the sequence of partial sums and determine if the series converges.  ∑n=1∞1n(n+1)

joshyoung05M · Accepted Answer

Given: The series ∑n=1∞1n(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=1∞1n(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(1n−1n+1)  ⇒tk=(11−12)+(12−13)+(13−14)+...+(1k−1k+1)  ⇒tk=(1−1k+1)  Hence general term of sequence of partial sum of this series is tk=(1−1k+1)  So sequence of partial sum is {1−1n+1}n=1∞  Now it is known that ∑n=1∞an=limk→∞∑n=1kan  Therefore here   ∑n=1∞1n(n+1)=limk→∞∑n=1k1n(n+1)  ⇒∑n=1∞1n(n+1)=limk→∞(1−1k+1)  ⇒∑n=1∞1n(n+1)=1  Hence the series is convergent as converges to 1.  Answer:  The sequence of partial sum is {1−1n+1}n=1∞  And the series is convergent as converges to 1.

Jeffrey Jordon · Accepted Answer

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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)}

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