Q.178. Series Summation

Question 178.
Find the sum of the following series.

Answer 178.

There are 99 terms.
nth term = 1 / [(n+1)√n + n√(n+1)]

Rationalising,
= [(n+1)√n - n√(n+1)] / { [(n+1)√n + n√(n+1)] * [(n+1)√n - n√(n+1)] }
= [(n+1)√n - n√(n+1)] / [(n+1)^2 * n - n^2 * (n+1)]
= [(n+1)√n - n√(n+1)] / [n(n + 1)]
= 1/√n - 1/√(n+1)
=>
t1 = 1 - 1/√2
t2 = 1/√2 - 1/√3
t3 = 1/√3 - 1/√4
:::::
:::::
tn = 1/√n - 1/√(n+1)
--------------------------------------…
Adding, Σ tn = 1 - 1/√(n+1)
Plugging n = 99
=> sum of the given series
= 1 - 1/√(99+1)
= 1 - 1/10
= 0.9.

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