#MAT187 PCE

Geometric Series

A geometric series is a series in which each term is multiplied by a value, which is known as the ratio. That is, $$\frac{a_{n+1}}{a_{n}}=r,\forall n\in N$$A geometric series follows the form: $$\sum_{n=0}^\infty c_{0}r^n$$

Ratio Test for Geometric Series

The ratio test for a geometric series governs the convergence of a geometric series.
For $\sum _{n=0}^{\mathrm{\infty }}{a}_n$, where $a_n\ne 0$ $\mathrm{\forall }n\in N$, let $p=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|$. Then,

1. If $p<1$, then the series converges.
2. If $p>1$, then the series diverges.
3. If $p=1$, the ratio test is inconclusive.
   In this case, $p$ is the ratio $r$, and it can be used to find the radius of convergence.

For a geometric series in which $x$ is a part of the $p$ term, the Radius of Convergence is the domain of values of $|x-a|$ that allows the series to converge.

• Essentially, it is the distance from $a$ to $x$, i.e. the distance from the centre point to $x$.

Example: $\sum _{n=0}^{\mathrm{\infty }}(4x{)}^n$

For the series to converge, $4|x|<1$ must be true. Thus,

And therefore, the radius of convergence is $\frac{1}{4}$.

P-test

For a function that can be represented in the form of:

It converges if $p>1$, and diverges if $p\le 1$.

Singularity??

For a function $\frac{1}{p(x)}$ the closest root (real or complex) of $p(x)$ to the centre point is the radius of convergence (absolute value of it, so for a complex number, it is the modulus).