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