Calculus- sequences advice, tips?

I can't do the exercise on my own, really appreciate if anybody can give me tips, advice on how to master this topic... To be honest, I don't really get the picture of what it's all about...Calculus- sequences advice, tips?
This question is very very vague.





The most fundamental series is the geometric series


1/(1 鈭?x) = 1+x+x^2+x^3+...





You can prove this by defining:


S = 1+x+x^2+x^3+...


xS = x+x^2+x^3+...


xS -S = -1


S=1/(1-x)





You can integrate the geometric series to get:


- ln (1-x) = x+x^2/2 + x^3/3 + ... where ln is the natural log





You can verify the exponential expansion easy enough


exp(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...


This is the only infinite polynomial that every time you take the derivative you get the exact same polynomial. That's the definition of exp(x). It is a function that equals it's own derivative.





Since you know that sin(x) = - sin(-x) the series expansion must only be odd powers of x. And the 2nd derivative of sign must equal the negative of the original function.


sin(x) = x - x^3/3! + x^5/5! ...


Just take two derivatives of the right hand side and you'll get the negative of what you started with.





Similar argument for cos(x)





I hope that was your questionCalculus- sequences advice, tips?
I am not sure exactly what kind of help that you need, but I have two suggestions.


The first is that you may want to see if your local library offers free tutoring through the internet. Many libraries pay for their patrons to get free tutoring through a service such as tutor.com. If yours does, then you can use your library card to get one on one help with your problems


The second is you may want to try the site


http://tutorial.math.lamar.edu/Classes/C鈥?/a>


The person gives very good down-to-earth explanations.