Match the following infinite sums. List I List II (A) ∑ n = 1 ∞ ( 2 n − 1 ) 2 ( 2 n + 1 ) 2 n (p) 4 1 (B) ∑ n = 1 ∞ n ( n + 1 ) n − n 2 − 1 (q) 8 1 (C) ∑ n = 1 ∞ n 2 ( n + 1 ) 2 2 n

Question

Match the following infinite sums.

List I	List II
$(A) \sum_{n = 1}^{\infty} \frac{n}{( 2 n - 1 ) ^{2} ( 2 n + 1 ) ^{2}}$	$(p) \frac{1}{4}$
$(B) \sum_{n = 1}^{\infty} \frac{n - n ^{2} - 1}{n ( n + 1 )}$	$(q) \frac{1}{8}$
$(C) \sum_{n = 1}^{\infty} \frac{2 n + 1}{n ^{2} ( n + 1 ) ^{2}}$	$(r) 1$
$(D) \sum_{n = 1}^{\infty} \frac{1}{( n + n + 1 ) n ( n + 1 )}$	$(s) \frac{1}{2}$
	$(t) \infty$

TestHub · Accepted Answer

(A) ( 2 n − 1 ) 2 ( 2 n + 1 ) 2 n ​ = 8 1 ​ [ ( 2 n − 1 ) 2 1 ​ − ( 2 n + 1 ) 2 1 ​ ] , sum = 8 1 ​ . (B) The term equals n + 1 n ​ ​ − n n − 1 ​ ​ , which telescopes to 1. (C) n 2 ( n + 1 ) 2 2 n + 1 ​ = n 2 1 ​ − ( n + 1 ) 2 1 ​ , sum = 1. (D) The term equals n ​ 1 ​ − n + 1 ​ 1 ​ , sum = 1.. .

Mathematics - Sequence & Series Question with Solution | TestHub

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