The sum of the series (9/52 ⋅ 2 ⋅ 1)+(13/53 ⋅ 3 ⋅ 2)+(17/54 ⋅ 4 ⋅ 3)+ ldots to infinite terms, is

Question

The sum of the series (9/52 ⋅ 2 ⋅ 1)+(13/53 ⋅ 3 ⋅ 2)+(17/54 ⋅ 4 ⋅ 3)+ ldots to infinite terms, is (A) (2/5) (B) (1/5) (C) 1 (D) None of these

Tardigrade Admin · Accepted Answer

The general term of the series is tr=(4 r+1/5r ⋅ r(r-1)), where r ≥ z =(5 r-(r-1)/5r ⋅ r(r-1))=(1/5r-1(r-1))-(1/5r ⋅ r) ∴ displaystyle∑r=2∞ tr=((1/51 ⋅ 1)-(1/52 ⋅ 2))+((1/52 ⋅ 2)-(1/53 ⋅ 3))+((1/53 ⋅ 3)-(1/54 ⋅ 4))+ â¦ to infinity =(1/5) (∵ terms tend to zero as n arrow ∞)

Q. The sum of the series $\frac{9}{5 ^{2} \cdot 2 \cdot 1} + \frac{13}{5 ^{3} \cdot 3 \cdot 2} + \frac{17}{5 ^{4} \cdot 4 \cdot 3} + \dots$ to infinite terms, is

$\frac{2}{5}$

$\frac{1}{5}$

$1$

None of these

Q. The sum of the series 52⋅2⋅19​+53⋅3⋅213​+54⋅4⋅317​+… to infinite terms, is

52​

51​

1

None of these

The general term of the series is tr​=5r⋅r(r−1)4r+1​, where r≥z =5r⋅r(r−1)5r−(r−1)​=5r−1(r−1)1​−5r⋅r1​ ∴r=2∑∞​tr​=(51⋅11​−52⋅21​)+(52⋅21​−53⋅31​)+(53⋅31​−54⋅41​)+… to infinity =51​ (∵ terms tend to zero as n→∞)

Q. The sum of the series $\frac{9}{5 ^{2} \cdot 2 \cdot 1} + \frac{13}{5 ^{3} \cdot 3 \cdot 2} + \frac{17}{5 ^{4} \cdot 4 \cdot 3} + \dots$ to infinite terms, is

$\frac{2}{5}$

$\frac{1}{5}$

$1$