Geometric Series

Key Questions

• How do you find the common ratio of an infinite geometric series?

  You can find the common ratio `r` by finding the ratio between any two consecutive terms.

  `r=a_1/a_0=a_2/a_1=a_3/a_2=cdots=a_{n+1}/a_n=cdots`

  For the geometric series `sum_{n=0}^infty(-1)^n{2^{2n}}/5^n`, the common ratio

  `r=a_1/a_0={-2^2/5}/{1}=-4/5`

  Wataru · · Sep 24 2014
• How do you find the sum of the geometric series `8+4+2+1+…`?

  The sum of a convergent geometric series `sum_{n=0}^{infty}ar^n` is `frac{a}{1-r}`. Since `a=8` and `r=1/2` in the posted geometric series, the sum is `frac{8}{1-frac{1}{2}}=16`.

  Wataru · · Aug 28 2014
• What is a Geometric Series?

  It's the sum of a sequence of terms where each term equals the previous one multiplied by a given ratio (e.g. `1 + 2 + 4 + 8...`).

  Eddie W. · 1 · Jul 30 2014

• What is a Geometric Series?
• How do you find `a_1` for the geometric series with `r=3` and `s_6=364`?
• How do you find the sum of finite geometric series?
• How do you find the sum of the geometric series `8+4+2+1+…`?
• How do you use a geometric series to prove that `0.999…=1`?
• What is `s_n` of the geometric series with `a_1=4`, `a_n=256`, and `n=4`?
• What is the formula for the sum of an infinite geometric series?
• How do you know when to use the geometric series test for an infinite series?
• How do you know when a geometric series converges?
• How do you find the sum of the infinite geometric series with `a_1=-5` and `r=1/6`?
• How do you find the common ratio of an infinite geometric series?
• How do you Use an infinite geometric series to express a repeating decimal as a fraction?
• What is the formula to find the sum of an infinite geometric series?
• What is the sum of the infinite geometric series with `a_1=42` and `r=6/5`?
• What is the sum of the infinite geometric series `sum_(n=1)^oo(1/3)^n` ?
• What is the sum of the infinite geometric series `sum_(n=0)^oo(1/e)^n` ?
• What is the sum of the infinite geometric series `sum_(n=1)^oo2^n/5^(n-1)` ?
• What is the sum of the infinite geometric series `1/8-1/4+1/2-1+…` ?
• What is the sum of the infinite geometric series `sum_(n=1)^oo6(0.9)^(n-1)` ?
• `lim_(n->oo)(1+1/n)^n =` ?
• `y_n=log x_n, n =2,3,4,...and y_n-(n-1)/n y_(n-1)=1/n log n`, with `y_2=log sqrt2`, how do you prove that `x_n=(n!)^(1/n)`?
• Question #57453
• How to calculate this limit?`lim_(n->oo)sum_(k=1)^n(1/2^k+1/3^k)`
• Sum of first four terms of a geometric series is `15` and next four terms is `240`. Find the first term and common ratio of the series?
• Question #38962