Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Series question, converge/diverge

Have no idea what to do. Help?

Original post by Airess3

Have no idea what to do. Help?

Note that

\tan\left(\frac{\pi}{4}+\frac{n \pi}{2}\right)\text{ is equal to }\pm1\text{ according as }n\text{ is even or odd}

}
Does this help ?

OP

Original post by brianeverit

Note that

\tan\left(\frac{\pi}{4}+\frac{n \pi}{2}\right)\text{ is equal to }\pm1\text{ according as }n\text{ is even or odd}

}
Does this help ?

So plug in 0 where all the n's are ? n is odd so in the brackets is -1?? But tan Pi/4=1? I'm confused. Do i have to take the limit of the equation?

You know from the post above that

\displaystyle {tan(\frac{\pi}{4}+\frac{n\pi}{2})}=(-1)^n

that is that the top alternates between

\displaystyle \pm 1

depending on whether

\displaystyle n

is odd or even. Indeed Brian's post says this.

So you can write the summand as

\displaystyle \frac{tan(\frac {\pi}{4}+\frac{n\pi}{2})}{\sqrt{1+\sqrt{n}}}=\frac {(-1)^n}{\sqrt{1+\sqrt{n}}}

Now this looks to me like a alternating sequence test problem. Are you familiar with this test for convergence and the necessary conditions for it's application? If so then you should be able to spot and prove (or state depending on how rigorous the proof needs to be) that these conditions are met and that this test is applicable then you are done.

(edited 9 years ago)

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