Question

For the following telescoping series, find a formula for the nth term of the sequence of partial sums ($S_n$). Then evaluate $\lim_{n \to \infty} S_n$ to obtain the value of the series or state that the series diverges. $\sum_{k=1}^{\infty} \left( \frac{4}{\sqrt{k+5}} - \frac{4}{\sqrt{k+7}} \right)$ $S_n = \frac{2\sqrt{6}}{3} - \frac{4\sqrt{7}}{7} - \left( \frac{4}{\sqrt{n+5}} - \frac{4}{\sqrt{n+7}} \right)$ (Type an exact answer, using radicals as needed.) Select the correct choice and fill in any answer boxes in your choice below. A. $\sum_{k=1}^{\infty} \left( \frac{4}{\sqrt{k+5}} - \frac{4}{\sqrt{k+7}} \right) = \boxed{}$ (Type an exact answer, using radicals as needed.) B. The series diverges.

          For the following telescoping series, find a formula for the nth term of the sequence of partial sums ($S_n$). Then evaluate $\lim_{n \to \infty} S_n$ to obtain the value of the series or state that the series diverges.
$\sum_{k=1}^{\infty} \left( \frac{4}{\sqrt{k+5}} - \frac{4}{\sqrt{k+7}} \right)$
$S_n = \frac{2\sqrt{6}}{3} - \frac{4\sqrt{7}}{7} - \left( \frac{4}{\sqrt{n+5}} - \frac{4}{\sqrt{n+7}} \right)$
(Type an exact answer, using radicals as needed.)
Select the correct choice and fill in any answer boxes in your choice below.
A. $\sum_{k=1}^{\infty} \left( \frac{4}{\sqrt{k+5}} - \frac{4}{\sqrt{k+7}} \right) = \boxed{}$ 
(Type an exact answer, using radicals as needed.)
B. The series diverges.

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.
∑k=1^∞( (4)/(√(k+5)) - (4)/(√(k+7)))
Sn = (2√(6))/(3) - (4√(7))/(7) - ( (4)/(√(n+5)) - (4)/(√(n+7)))
(Type an exact answer, using radicals as needed.)
Select the correct choice and fill in any answer boxes in your choice below.
A. ∑k=1^∞( (4)/(√(k+5)) - (4)/(√(k+7))) =
(Type an exact answer, using radicals as needed.)
B. The series diverges.

Added by Arthur L.

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