Provide an appropriate response.
-Which of the following statements is false?
A) If $\mathrm { a } _ { \mathrm { n } }$ and $\mathrm { f } ( \mathrm { n } )$ satisfy the requirements of the Integral Test, and if $\int _ { 1 } ^ { \infty } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ converges, then $\sum _ { \mathrm { n } = 1 } ^ { \infty } \mathrm { a } _ { \mathrm { n } } =$ $\int _ { 1 } ^ { \infty } f ( x ) d x$
B) The integral test does not apply to divergent sequences.
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { { } _ { n } p }$ converges if $p 1$ and diverges if $p \leq 1$.
D) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) p }$ converges if $p 1$.

Question

Provide an appropriate response.
-Which of the following statements is false?
A) If $\mathrm { a } _ { \mathrm { n } }$ and $\mathrm { f } ( \mathrm { n } )$ satisfy the requirements of the Integral Test, and if $\int _ { 1 } ^ { \infty } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ converges, then $\sum _ { \mathrm { n } = 1 } ^ { \infty } \mathrm { a } _ { \mathrm { n } } =$ $\int _ { 1 } ^ { \infty } f ( x ) d x$
B) The integral test does not apply to divergent sequences.
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { { } _ { n } p }$ converges if $p > 1$ and diverges if $p \leq 1$.
D) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) p }$ converges if $p > 1$.

Quizplus · Accepted Answer

All the other statements are true. The integral test can only be applied to series with non-negative terms, which converge if and only if the corresponding improper integral converges. In statement A, the condition that $\int _ { 1 } ^ { \infty } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ converges is necessary, but not sufficient, for the convergence of $\sum _ { \mathrm { n } = 1 } ^ { \infty } \mathrm { a } _ { \mathrm { n } }$ to be established.

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Provide an Appropriate Response $\mathrm { a } _ { \mathrm { n } }$