Evaluate 1/2 − 1/ ( 2 ⋅ 2 2 ) + 1/ ( 3 ⋅ 2 3 ) − 1/ ( 4 ⋅ 2 4 ) + ⋯ .

The Taylor series ln ( 1 + t ) = ∑ k = 1 ∞ ( − 1 ) k + 1 t k / k at t = 1/2 gives ln ( 3/2 ) .

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Evaluate $1/2 - 1/ (2 \cdot 2^{2}) + 1/ (3 \cdot 2^{3}) - 1/ (4 \cdot 2^{4}) + \dots$ .

Answer:

Solution:

The Taylor series $ln (1 + t) = \sum_{k = 1}^{\infty} (- 1)^{k + 1} t^{k} / k$ at $t = 1/2$ gives $ln (3/2)$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Special sequences/series

⏱ 2mℹ️ Source: QB

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