Answer to Question #212285 in Calculus for Vikas

Question #212285

show that∑n=0∞ (-1)^(n+1)(5)/(7n+2)` is conditionally convergent 


1
Expert's answer
2021-07-16T02:38:08-0400

Consider the series


"\\displaystyle\\sum_{n=0}^{\\infin}|\\dfrac{(-1)^{n+1}}{7n+2}|=\\displaystyle\\sum_{n=0}^{\\infin}\\dfrac{1}{7n+2}"

Use the Integral Test

Let "f(n)=a_n=\\dfrac{1}{7n+2}, n=0,1,2,..."


"\\displaystyle\\int_{0}^{\\infin}\\dfrac{1}{7x+2}dx=\\lim\\limits_{t\\to\\infin}\\displaystyle\\int_{0}^{t}\\dfrac{1}{7x+2}dx"

"=\\lim\\limits_{t\\to\\infin}\\dfrac{1}{7}[\\ln|7x+2|]\\begin{matrix}\n t \\\\\n 0\n\\end{matrix}=\\infin"

Since this improper integral is divergent, the series "\\displaystyle\\sum_{n=0}^{\\infin}\\dfrac{1}{7n+2}" diverges by the Integral Test.

Consider the alternating series


"\\displaystyle\\sum_{n=0}^{\\infin}\\dfrac{(-1)^{n+1}}{7n+2}""b_n=\\dfrac{1}{7n+2}>0, n=0,1,2,..."

"b_{n+1}=\\dfrac{1}{7(n+1)+2}<\\dfrac{1}{7n+2}=b_n, n=0, 1, 2, ..."

"\\lim\\limits_{n\\to \\infin}b_n=\\lim\\limits_{n\\to \\infin}\\dfrac{1}{7n+2}=0"

Then the series "\\displaystyle\\sum_{n=0}^{\\infin}\\dfrac{(-1)^{n+1}}{7n+2}" is convergent by the Alternating Series Test.


Therefore the series "\\displaystyle\\sum_{n=0}^{\\infin}\\dfrac{(-1)^{n+1}}{7n+2}" is conditionally convergent.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS