Question

Convergence Tests for Series: Fill in the blanks.

The Integral Test: If $a\left(x\right)$is a function that is ___, ___, and ___ on $\left[1,\infty \right)$and $\left\{a_k\right\}$is the sequence defined by $a_k=\_\_\_$for every $k\in Z^{+}$, then $\sum _{k=1}^{\infty }{a}_k$and role="math" localid="1649357478850" ${\int }_1^{\infty }a\left(x\right)dx$either both ___ or both ___.

Short answer

If $a\left(x\right)$is a function that is positive, decreasing, and continuous on $\left[1,\infty \right)$and is the sequence defined by $a_k=a\left(k\right)$for every $k\in Z^{+}$, then $\sum _{k=1}^{\infty }{a}_k$and ${\int }_1^{\infty }a\left(x\right)dx$ either both converge or both diverge.

Step-by-step solution

Step 1. Given Information

The test is the integral test.

Step 2. Explanation.

• The integral test is applicable if there exists a function $a\left(x\right)$such that role="math" localid="1649357829156" $a\left(k\right)=a_k$which is positive, decreasing and continuous function.
• The function must be defined on the interval $\left[1,\infty \right)$.
• If such a function exist then$\sum _{k=1}^{\infty }{a}_k$converges or diverges if${\int }_1^{\infty }a\left(x\right)dx$converges or diverges.