Question

54–69. Telescoping seriesFor the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

Accepted Answer

Start by factoring the denominator of the general term in the series. The denominator is $$k^2 + 5k + 4. $$Factor it as $$(k + 1)(k + 4). $$Rewrite the general term $$\frac{3}{k^2 + 5k + 4} as$$ $$\frac{3}{(k + 1)(k + 4)}. $$Next, use partial fraction decomposition to express this as $$\frac{A}{k + 1} + \frac{B}{k + 4}$$ for constants $$A$$ and $$B. $$Set up the equation $$\frac{3}{(k + 1)(k + 4)} = \frac{A}{k + 1} + \frac{B}{k + 4}. $$Multiply both sides by $$(k + 1)(k + 4) to $$get $$3 = A(k + 4) + B(k + 1). $$Solve for $$A$$ and $$B by $$choosing convenient values for $$k or by $$equating coefficients. Once you find $$A$$ and $$B$$, write the nth partial sum $$S_n = \sum_{k=1}^n \frac{3}{(k + 1)(k + 4)} = \sum_{k=1}^n \left( \frac{A}{k + 1} + \frac{B}{k + 4} ight). $$Separate the sums and write out the terms to identify the telescoping pattern. Use the telescoping property to cancel terms and find a simplified expression for $$S_n. $$Then, evaluate $$\lim_{n 	o \infty} S_n by $$analyzing the behavior of the remaining terms as $$n$$ approaches infinity to determine if the series converges or diverges.

54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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Limit of Partial Sums and Convergence