Geometric Series

Understanding geometric series requires examining multiple perspectives and considerations. Proof of geometric series formula - Mathematics Stack Exchange. Proof of geometric series formula Ask Question Asked 4 years, 1 month ago Modified 4 years, 1 month ago Calculate expectation of a geometric random variable. 2 A clever solution to find the expected value of a geometric r.v.

is those employed in this video lecture of the MITx course "Introduction to Probability: Part 1 - The Fundamentals" (by the way, an extremely enjoyable course) and based on (a) the memoryless property of the geometric r.v. and (b) the total expectation theorem. How to Recognize a Geometric Series - Mathematics Stack Exchange.

The definition of a geometric series is a series where the ratio of consecutive terms is constant. It doesn't matter how it's indexed or what the first term is or whether you have a constant. When is a Power Series a Geometric Series? - Mathematics Stack Exchange. So surely you see the answer now, but I'll state it for the record: a power series is a geometric series if its coefficients are constant (i.e. In particular, not all power series are geometric.

For example $\sum x^n$ is geometric, but $\sum \frac {x^n} {n!}$ is not. Alternative proofs of convergence of geometric series. The first way of handling geometric series I saw, was to derive the closed form for the finite sum. In relation to this, discovering this derivation seems pretty reasonable if one stares at the geometric sum long enough: How to derive the closed form solution of geometric series. Series expansion: $\\frac{1}{(1-x)^n}$ - Mathematics Stack Exchange.

What is the expansion for $(1-x)^{-n}$? Could find only the expansion upto the power of $-3$. Is there some general formula? Starting index for geometric series test - Mathematics Stack Exchange. 1 I was just wondering if the geometric series test for series of the form $\sum_ {n=}^ {\infty}ar^ {n}$ needs the index to start at $0$ or $1$?

From my understanding of the proof using partial sums, calculating the convergent value as $\frac {a} {1-r}$ requires the series to start at $0$. Geometric series of matrices - Mathematics Stack Exchange. Geometric series of matrices Ask Question Asked 11 years, 4 months ago Modified 11 years, 4 months ago statistics - Proof variance of Geometric Distribution - Mathematics ....

I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success. The distribution function is $P (X=x) = q^x p$ for $x=0,1,2,\ldots$ and $q = 1-p$.