Question:
The value of
\[
2 + \frac{1}{5} + \frac{1}{3} + \frac{1}{5^3} + \frac{1}{5^5} + \dots
\]
is:
The value of
\[
2 + \frac{1}{5} + \frac{1}{3} + \frac{1}{5^3} + \frac{1}{5^5} + \dots
\]
is:
Show Hint
The sum of a geometric series can be found using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
Updated On: Apr 22, 2026
- \( \log 2 + \log 3 \)
- \( \log 2 + 2 \)
- \( \frac{1}{2} \log 2 \)
- \( \log 3 \)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Identify the series.
The given series is a standard series with powers of 5. Recognizing this as a geometric series, we write: \[ S = 2 + \sum_{n=1}^{\infty} \frac{1}{5^{2n-1}} = 2 + \sum_{n=1}^{\infty} \frac{1}{5^{2n-1}} = 2 + \frac{1}{5} + \frac{1}{5^3} + \cdots \]
Step 2: Use the geometric series formula.
The sum of an infinite geometric series \( a + ar + ar^2 + \cdots \) is: \[ S = \frac{a}{1 - r}, \quad |r|<1 \] For our series, \( a = 2 \) and \( r = \frac{1}{5^2} \).
Step 3: Calculate the sum of the series.
The sum of the series is: \[ S = 2 + \frac{2}{1 - \frac{1}{25}} = 2 + \frac{2}{\frac{24}{25}} = 2 + \frac{50}{24} \]
Step 4: Final result.
Simplify the result: \[ S = 2 + 2 = \log 2 + 2 \] Thus, the correct answer is option (B).
The given series is a standard series with powers of 5. Recognizing this as a geometric series, we write: \[ S = 2 + \sum_{n=1}^{\infty} \frac{1}{5^{2n-1}} = 2 + \sum_{n=1}^{\infty} \frac{1}{5^{2n-1}} = 2 + \frac{1}{5} + \frac{1}{5^3} + \cdots \]
Step 2: Use the geometric series formula.
The sum of an infinite geometric series \( a + ar + ar^2 + \cdots \) is: \[ S = \frac{a}{1 - r}, \quad |r|<1 \] For our series, \( a = 2 \) and \( r = \frac{1}{5^2} \).
Step 3: Calculate the sum of the series.
The sum of the series is: \[ S = 2 + \frac{2}{1 - \frac{1}{25}} = 2 + \frac{2}{\frac{24}{25}} = 2 + \frac{50}{24} \]
Step 4: Final result.
Simplify the result: \[ S = 2 + 2 = \log 2 + 2 \] Thus, the correct answer is option (B).
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