ok i did a couple of problems and i was wondering if someone could tell me if i did them right? the first one is on convergence and the second which was a little more involved asked me to find the 3rd degree taylor polynomial...thanks in advanceIMG.pdf
For the second question, assume that you can write as a polynomial centred at (I assume that's what you meant).
Therefore
.
At , you find .
Differentiate both sides
.
At , you find
.
Differentiate both sides
.
At you find
.
Differentiate both sides
.
At you have
.
Can you finish the last step to find ?
Also, to answer your question, no the ratio test is not the only way to prove convergence. It IS however an EASY way to prove convergence.
The reason you got the wrong answer is because you made the mistake I pointed out at the beginning of the post.