The value of ∑ n = 1 ∞ ( − 1 ) n + 1 5 n n is b a , where a and b are relatively prime. Then

S = 5 1 − 5 2 2 + 5 3 3 − … and − 5 S = − 5 2 1 + 5 3 2 − … Subtracting the second line from the first, S + 5 S = 5 6 S = 5 1 − 5 2 1 + 5 3 1 − ⋯ = 1 + 1/5 1/5 = 6 1 . So S = 36 5 . Here a = 5 , b = 36 : a 3 , b > 10 .

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Mathematics - Sequence & Series Question with Solution | TestHub

MathematicsSequence & SeriesA.G.P.Medium2 minQB

MathematicsMediummultiple choice

The value of $\sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{n}{5 ^{n}}$ is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Then

Options:(select one or more)

Answer:

B, C, D

Solution:

$S = \frac{1}{5} - \frac{2}{5 ^{2}} + \frac{3}{5 ^{3}} - \dots$ and $- \frac{S}{5} = - \frac{1}{5 ^{2}} + \frac{2}{5 ^{3}} - \dots$ Subtracting the second line from the first, $S + \frac{S}{5} = \frac{6 S}{5} = \frac{1}{5} - \frac{1}{5 ^{2}} + \frac{1}{5 ^{3}} - \dots = \frac{1/5}{1 + 1/5} = \frac{1}{6}$ . So $S = \frac{5}{36}$ . Here $a = 5, b = 36 : a < b, a > 3, b > 10$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:A.G.P.

⏱ 2mℹ️ Source: QB

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