Hey guys I feel quite silly asking this but I'm quite confused about finding the radius of convergence, I've looked up video's on youtube, notes online and of course my own notes but without a clear example I'm lost. The question I'm working on, I believe that it's supposed to be quite easy as it's only worth 5 marks. If anyone can help/explain I'd really appreciate it:
find the radius of convergence of the series $\sum_{n=0}^\infty 10^nz^n $
Thanks in advance!
Do you know the root test? Try it on the series $$ \sum_{n=0}^\infty 10^n z^n $$ First we investigate convergence of the sequence $$ \big|10^n z^n\big|^{1/n} = 10 |z|. $$ It converges to $10 |z|$. Easy. Now we look up what that means. If $10 |z| < 1$, the series converges. If $10 |z| > 1$ the series diverges. [If $10 |z| = 1$ the test is inconclusive, but for the radius of convergence we do not care about that.] So: $$ |z| < \frac{1}{10},\qquad\text{converges} \\ |z| > \frac{1}{10},\qquad\text{diverges}. $$ The "radius of convergence" is $1/10$.
You might rewrite your series as $$ \sum_{n = 0}^\infty (10z)^n,$$ in which case you would recognize it as a geometric series. You could also apply the ratio test or the root test from first year calculus.
Set $u=10z$. This becomes the geometric series $\sum u^n$, which has radius of convergence $10z=u<1$. Hence for $z$, the radius of convergence is $\frac1{10}$.
The geometric series converges when $|r|<1$.