$\text { Determine the convergence or divergence of the series } \frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 } \ldots$
A) The series $\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }$.
diverges.
B) The series $\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }$
converges.

Question

$\text { Determine the convergence or divergence of the series } \frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 } \ldots$ 
A) The series $\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }$.
diverges.
B) The series $\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }$
converges.

Quizplus · Accepted Answer

The terms of the series are positive, so we can apply the Integral Test for convergence. Suppose $f(x)=\frac{1}{x}.$ Then, $f$ is decreasing on $(1,\infty)$ and
$$\int_{250}^{\infty} f(x) \, dx=\ln(250)-\lim_{t \to \infty} \ln(t)=\ln\left(\frac{250}{\infty}\right)= \ln 0= -\infty.$$
Since the integral diverges, the series diverges by the Integral Test. Therefore, the correct answer is $\boxed{\textbf{(B)}}.$

A) the Series 1251+1258+1277+1314\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }2511​+2581​+2771​+3141​

A) the Series $\frac { 1 } { 251 } + \frac { 1 } { 258 } + \frac { 1 } { 277 } + \frac { 1 } { 314 }$