Question

Use the first three non-zero terms of the Maclaurin series to approximate the value of the definite integral

Answer 1

\[\int\limits_{0}^{0.8} (1-cost)/t^2 dt\]

Answer 2

you know the power series for cosine?

Answer 3

And find an appropriate test to determine whether or not the series converges

Answer 4

I forget :S

Answer 5

man this is a pain. last one for me i think
ok for cosine you only need to remember two things:
\(\cos(0)=1\) so expansion starts with 1
and \(\cos(x)\) is even so you only have even terms.
therefore
\[\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+...\]

Answer 6

subtract this mess from one and get
\[\frac{x^2}{2}-\frac{x^4}{4!}+\frac{x^6}{6!}-...\]

Answer 7

so far so good?

Answer 8

Yup!

Answer 9

now i guess we have to divide by \(x^2\) but that is easy enough. we get \[\frac{1}{2}-\frac{x^2}{24}+\frac{x^4}{720}\]three terms is apparently enough

Answer 10

lord now we do what? take the anti derivative and plug in .8 and 0 and subtract. well at 0 we will get zero, but i have no idea what you get at .8

Answer 11

again the anti derivative should be ok right?

Answer 12

and the rest is an annoying calculator exercise whcih will leave up to you

Answer 13

Thanks! Would you know what to use for this? find an appropriate test to determine whether or not the series converges

Answer 14

series converges for sure, but i am not sure what test you are supposed to use. what do you have available?

Answer 15

@Zarkon what does it mean "use appropriate test" for this one, any idea?

Answer 16

Root test, ratio test, condensation test

Answer 17

use a ratio test