Question

confirm the integral test for sum n=1 to infinity 1/2^x

Answer 1

\[\int_{1}^{\infty}(\frac12)^xdx\]\[u=(\frac12)^x\]\[du=(\frac12)^x\ln (\frac12)dx=-\ln 2(\frac12)^xdx\implies(\frac12)^xdx=-\frac{du}{\ln2}\]

Answer 2

can you do the improper integral now?

Answer 3

i havent read it yet but i assume its just like regular integration, in the problem i posted on the 2 is raised to the x power

Answer 4

do you know how to convert improper integrals to limit-integral pairs?

Answer 5

no i ont i just started the topic

Answer 6

then you should read this
http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx
but I will set it up for you this time

Answer 7

thanks

Answer 8

\[\int_{1}^{\infty}(\frac12)^xdx=\lim_{n\to\infty}\int_{1}^{n}(\frac12)^xdx\]I showed you the u-substitution here gives\[-{1\over\ln2}\int du=-{u\over\ln2}\]putting in the bounds, and making the bound at infinity a limit as n goes to infinity, our sub gives us\[\lim_{n\to\infty}{1\over\ln2}(\frac12)^x|_{n}^{1}\]evaluate and take the limit

Answer 9

thanks

Answer 10

welcome