The sum of the first n terms of the series 1^2-2^2+3^2-4^2+5^2-6^2+…is (a) (n(n+1))/(2) (b) ((-1

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The sum of the first n terms of the series...

The sum of the first $\mathrm{n}$ terms of the series $1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-6^{2}+\ldots$ is
(a) $\frac{\mathrm{n}(\mathrm{n}+1)}{2}$
(b) $\frac{(-1)^{m-1} \mathrm{n}(\mathrm{n}+1)}{2}$
(c) $\frac{-n(n+1)}{2}$
(d) $\mathrm{n}(\mathrm{n}+1)$

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Key Concepts

Parity Considerations

Parity considerations refer to the analysis of whether a term's position in a series (odd or even) influences its contribution to the sum. In alternating series, this is crucial because the sign assigned to each term may depend on its parity, thereby shaping the overall expression for the sum.

Alternating Series

An alternating series is a sequence where the signs of consecutive terms alternate between positive and negative. This characteristic requires careful treatment when summing the series since positive and negative terms may cancel out, affecting convergence and the overall sum.

Series Summation Techniques

Series summation techniques involve methods for finding a closed-form expression for the sum of a series. Common approaches include grouping or pairing terms, exploiting patterns, using telescoping sums, and applying formulae for well-known series, all critical for efficiently evaluating complex series.

Polynomial Sequences

When a series involves polynomial expressions, such as squares or higher powers, the inherent patterns of polynomial growth must be considered. Summing such sequences often uses established formulae for the sums of powers, which might be adjusted when the series includes alternating signs.

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