what is the interval of convergence and how would you find it in an infinite geometric series?

Question

amistre64 · Answer

the interval of convergence is the interval on which a given polynomial matches a function that it is designed for

amistre64 · Answer

you find it by taking the limit of the ratio of next/now of the summation

anonymous · Answer

so if I had like 1 + x^2 + x^4... my interval would be x^2?

amistre64 · Answer

no, an interval is a span of the x axis for which the function and its power series actually match

amistre64 · Answer

x^2 is not an interval; its a monomial

anonymous · Answer

so how would I find it?

amistre64 · Answer

well, i already explained how you would find it; if you hd a specific example to run thru that might be better

anonymous · Answer

alright, how about 1 + (x-3) + (x-3)^2 +...

amistre64 · Answer

$$\sum_{0}^{inf} (x-3)^n\to \ \lim\frac{(x-3)^n}{(x-3)^{n-1}}=\lim x-3$$
pull out anything that has no "n" in it$$|x-3|\lim_{n \to inf}\ 1=|x-3|$$

amistre64 · Answer

the interval of convergence is when |x-3| < 1; or rather when |x| < 3

amistre64 · Answer

if i remember the material correctly

anonymous · Answer

ahaha I'm sure you do! Thank you so much!

amistre64 · Answer