# Solve for: sum from k=1 to+infinity of (6)/(sqrt(k^5))

## Expression: $\sum_{ k=1 }^{ +\infty } \frac{ 6 }{ \sqrt{ {k}^{5} } }$

Write the $n$-th term $\frac{ 6 }{ \sqrt{ {k}^{5} } }$ as a function

$f\left( x \right)=\frac{ 6 }{ \sqrt{ {x}^{5} } }$

Since the function $f\left( x \right)=\frac{ 6 }{ \sqrt{ {x}^{5} } }$ is positive, continuous, and decreasing for $x \geq 1$, use the Integral Test

$\int_{ 1 }^{ +\infty } \frac{ 6 }{ \sqrt{ {x}^{5} } } \mathrm{d} x$

To evaluate the improper integral, by definition, rewrite it using a limit and a definite integral

$\lim_{a \rightarrow +\infty} \left(\int_{ 1 }^{ a } \frac{ 6 }{ \sqrt{ {x}^{5} } } \mathrm{d} x\right)$

Evaluate the definite integral

$\lim_{a \rightarrow +\infty} \left(-\frac{ 4 }{ \sqrt{ a } \times |a| }+4\right)$

Evaluate the limit

$4$

Since the limit is finite, the improper integral converges

$\textnormal{Convergent}$

Since the integral converges, the given series also converges

$\textnormal{Converges}$

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