Question
Download Solution PDFWhat is the sum of the series \(1-\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{8}+...\) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Infinite Sum of the GP:
For a GP with common ratio r and first term a, the infinite sum is given by:
\(S_{\infty} = \dfrac{a}{1-r}\), Where \(\rm |r| < 1\).
Calculation:
Observe that the given series is a geometric progression with common ratio \(\rm r = -\dfrac{1}{2}\) and first term \(\rm a=1\).
Therefore, the infinite sum is:
\(\begin{align*} S_{\infty} &= \dfrac{1}{1-\left(-\frac{1}{2}\right)}\\ &= \dfrac{1}{1+\frac{1}{2}}\\ &= \dfrac{2}{3} \end{align*}\)
Therefore, the given sum is equal to 2/3.
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