Infinite Geometric Series as a Limit
Read this topicExplanations 2
- When it comes to infinite series, there are several very common types. Perhaps the most frequently occurring infinite series is a Geometric Series. Before discussing Geometric S…
By Victoria Rodriguez, Alex Federspiel
- The infinite geometric series ∑k=0∞ark=a+ar+ar2+ar3+…\sum_{k=0}^{\infty} ar^k = a + ar + ar^2 + ar^3 + \dots∑k=0∞ark=a+ar+ar2+ar3+… converges if ∣r∣<1|r|<1∣r∣<1, but it diverge…
By Victor Wang
Problems 6
- Let f(x)=1+12x+14x2+18x3+…f(x) = 1 + \frac12 x + \frac14 x^2 + \frac18 x^3 + \dotsf(x)=1+21x+41x2+81x3+… for −1≤x≤1-1\le x\le 1−1≤x≤1. Find e∫01f(x)dxe^{\int_0^1 f(x) dx}e∫01…
- What is the interval of convergence for the following series? ∑i=0∞(x2x+1)i\sum_{i=0}^\infty \left(\frac{x^2}{x+1}\right)^ii=0∑∞(x+1x2)i
- Using an infinite geometric series, what is the binary value 0.1010101…0.1010101\dots0.1010101… equal to in base 10? Express your answer as a fraction in lowest terms. (Hint: No…
- What is the value of 0.3003003003…0.3003003003\dots0.3003003003… as a fraction in lowest terms?
- Determine if the following series converges or diverges. If it converges, find its value ∑k=0∞3−k\sum_{k=0}^{\infty}3^{-k}k=0∑∞3−k
- Let nnn be any natural number that can be written as a product of powers of 222 and 333 (i.e. n=2x3yn = 2^x 3^yn=2x3y for non-negative integers xxx and yyy). What is the sum of …