Does the series $\sum _ { k = 1 } ^ { \infty } \frac { 5 } { k ^ { 2 } + 3 k + 2 }$ converge or diverge? If it converges, then find the sum.
A) Diverges
B) Converges, sum is 5/2
C) Converges, sum is 5/3
D) Converges, sum is 2/3

Question

Quizplus · Accepted Answer

The given series is $\sum _ { k = 1 } ^ { \infty } \frac { 5 } { k ^ { 2 } + 3 k + 2 }$. First, factor the denominator: $k^2 + 3k + 2 = (k + 1)(k + 2)$. Thus, the series can be rewritten as $\sum _ { k = 1 } ^ { \infty } \frac { 5 } { (k + 1)(k + 2) }$. This can be further decomposed using partial fraction decomposition: $\frac{5}{(k+1)(k+2)} = \frac{A}{k+1} + \frac{B}{k+2}$. Solving for A and B, we find that $A = 5$ and $B = -5$, so the series becomes $\sum _ { k = 1 } ^ { \infty } ( \frac{5}{k+1} - \frac{5}{k+2} )$. This is a telescoping series, where most terms cancel out, leaving only the first few and last few terms. Specifically, the first term is $5/2$ and the last term tends to 0 as $k$ approaches infinity. Therefore, the sum of the series converges to $5/2$.

Does the Series ∑k=1∞5k2+3k+2\sum _ { k = 1 } ^ { \infty } \frac { 5 } { k ^ { 2 } + 3 k + 2 }∑k=1∞​k2+3k+25​

Does the Series $\sum _ { k = 1 } ^ { \infty } \frac { 5 } { k ^ { 2 } + 3 k + 2 }$