# Geometric Series

A geometric series is a fundamental concept in mathematics, particularly in the realm of sequences and series. It comprises a sequence of numbers where each term is found by multiplying the preceding term by a constant factor, known as the common ratio. This series is characterized by its distinct pattern of growth or decay, making it a vital tool in various mathematical and real-world applications. Understanding the properties and behavior of geometric series is crucial in fields such as finance, physics, and engineering, where exponential growth or decay phenomena are prevalent.

Questions

- How do you know when a geometric series converges?
- How do you find the sum of the geometric series #8+4+2+1+…#?
- How do you find the sum of the infinite geometric series with #a_1=-5# and #r=1/6#?
- What is the formula for the sum of an infinite geometric series?
- What is the sum of the infinite geometric series #sum_(n=0)^oo(1/e)^n# ?
- How do you find #a_1# for the geometric series with #r=3# and #s_6=364#?
- What is #s_n# of the geometric series with #a_1=4#, #a_n=256#, and #n=4#?
- Show that #lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=9/40 a^(-31/24)#?
- How do you Use an infinite geometric series to express a repeating decimal as a fraction?
- 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?
- # lim_(n->oo)(1+1/n)^n = # ?
- How do you use a geometric series to prove that #0.999…=1#?
- How do you solve the series (7 - 9 + (81/7) - (729/49) + ... )?
- Show that lim x->a (x^3/8-a^3/8)/(x^5/3-a^5/3)?
- How do you find the sum of finite geometric series?
- What is the formula to find the sum of an infinite geometric series?
- #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)#?
- How do you find the common ratio of an infinite geometric series?
- 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)^oo2^n/5^(n-1)# ?