Infinite Geometric Series and the Geometric Series Test
Read this topicExplanations 10
- A geometric series is the sum of the terms of a geometric sequence. Let a geometric sequence be: a,ar,ar2,ar3,...a, ar, ar^2, ar^3, ...a,ar,ar2,ar3,... S=a+ar+ar2+ar3+...arn−1S …
By Michael Yang, Ruiran Xun, Anusha Rahman
- If we have a sequence such as 12+14+18+116+…\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots21+41+81+161+…, which never stops, our sequence is called an infini…
By Vivian Loh, Cameron Montag, Anusha Rahman
- (Video) Geometric Series and the Test for Divergence by PatrickJMT The video above by PatrickJMT explains how to identify a Geometric series and test it for series divergence or…
By Stefan Cuevas, Alex Federspiel, Anusha Rahman
- (Video) Geometric Series by Mathispower4u The video above by Mathispower4u explains how to determine if a Geometric series converges or diverges.
By Stefan Cuevas, Alex Federspiel
- (Video) Sum of an Infinite Geometric Series by PatrickJMT In the above video by PatrickJMT examples of finding the sum of geometric series are given.
By Stefan Cuevas, Alex Federspiel
- (Video) Writing a Geometric Series Using Sigma Notation by PatrickJMT In the above video by PatrickJMT it is shown how to write geometric series in sigma notation.
By Stefan Cuevas, Alex Federspiel
- (Video) Infinite Geometric Series Finding the Sum by The Math Sorcerer In the above video by The Math Wizard it is shown how to find the sum of a geometric series.
By Stefan Cuevas, Alex Federspiel
- (Video) The Super Formula for Infinite Geometric Series by The Math Sorcerer In the above video by The Math Sorcerer he explains what he calls the "super formula" for Infinite G…
By Stefan Cuevas, Alex Federspiel
- (Video) Limiting Sum of a Geometric Progression by Eddie Woo Here's a video by Eddie Woo where he explains what a limiting sum is and how you figure it out. Recall that the sum …
By Alex Federspiel, Anusha Rahman
- Infinite Geometric Series and the Geometric Series Test As we've learned previously, an infinite geometric series is a geometric series that continues forever. Each term is foun…
By Anusha Rahman
Problems 27
- What is the probability that Alice wins?
- If 1x+12x2+14x3+18x4+116x5+⋯=164,\frac{1} {x} + \frac{1} {2x^2} +\frac{1} {4x^3}+\frac{1}{8x^4}+\frac{1}{16x^5}+\cdots=\frac{1} {64},x1+2x21+4x31+8x41+16x51+⋯=641, and xxx…
- The sum of an infinite geometric series is a positive number SSS, and the second term in the series is 111. What is the smallest possible value of S?S?S? (2016 AMC 10B)
- What does the following series converge to? 1+110+1100+11000+…1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\dots1+101+1001+10001+…
- What does the following series converge to? 9+.9+.09+.009+…9+.9+.09+.009+\dots9+.9+.09+.009+…
- True or False\text{True or False}True or False === 0.9999999999⋯=10.9999999999\dots = 10.9999999999⋯=1
- Is the following series a geometric series? 5+25+125+…5 + 25 + 125 + \dots5+25+125+…
- Does 5+25+125+…5 + 25 + 125 + \dots5+25+125+… converge or diverge?
- What does the series 15+125+1125+⋯\frac{1}{5} + \frac{1}{25} + \frac{1}{125}+\cdots51+251+1251+⋯ converge to?
- Is the following series a geometric series? 3+9+27+…3 + 9 + 27 + \dots3+9+27+…
- Is the following series a geometric series? 3+6+9+…3 + 6 + 9 + \dots3+6+9+…
- Find the sum of the following infinite geometric series 9−3+1−13+…9-3+1-\frac{1}{3}+\dots9−3+1−31+…
- Find the sum of the following infinite geometric series: 12+38+932+27128+⋯\frac{1}{2}+\frac{3}{8}+\frac{9}{32}+\frac{27}{128}+\cdots21+83+329+12827+⋯
- Does the series −2+4+−8+16+…-2 + 4 + -8 + 16 + \dots−2+4+−8+16+… converge or diverge?
- Is the following series a geometric series? 5−25+125−625+…5 - 25 + 125 - 625 + \dots5−25+125−625+…
- Is the following series a geometric series? 3+6+9+12+⋯3 + 6 + 9 + 12 + \cdots3+6+9+12+⋯?
- What is the sum of the infinite geometric sequence 8,−4,2,−1,...8,-4,2,-1,...8,−4,2,−1,...?
- Is the following series a geometric series? 1+10+100+1000+…1+10+100+1000+\dots1+10+100+1000+…
- Suppose that the sum of a geometric series a+ar+ar2+ar3+⋯a + ar + ar^2 + ar^3 + \cdotsa+ar+ar2+ar3+⋯ is 999, while the sum of the terms in the series with odd powers of rrr is 4…
- Find the sum of the following series: 12+14−18+116+132−164+1128+1256−1512+⋯\frac12+\frac14-\frac18+\frac1{16}+\frac1{32} - \frac1{64} + \frac1{128} + \frac1{256} - \frac1{512} +…
- Find the sum of the series 15+252+153+254+155+256+⋯\frac15 + \frac2{5^2} + \frac1{5^3} + \frac2{5^4} + \frac1{5^5} + \frac2{5^6} + \cdots51+522+531+542+551+562+⋯.
- ∑i=1∞32i\sum_{i=1}^\infty \frac{3}{2^i}i=1∑∞2i3 If the above series is convergent, what does it converge to? If it does not converge, answer "No".
- ∑i=1∞(273274)i\sum_{i=1}^\infty \left(\frac{273}{274}\right)^ii=1∑∞(274273)i If the above series is convergent, what does it converge to?
- Suppose that, for some value of xxx, ∑i=1∞(373−2x)i\sum_{i=1}^\infty \left(\frac{37}{3} - 2x\right)^ii=1∑∞(337−2x)i converges. If xxx is an integer, find xxx.
- What is the value of 131+132+133+…\frac{1}{3^1} + \frac{1}{3^2} + \frac{1}{3^3} + \dots311+321+331+…? This series can also be written in summation notation as: ∑i=1∞13i\sum_{…
- A circle is inscribed in a square of area 444. Another square is inscribed in the circle. If this pattern goes on forever, what is the sum of the areas of the squares?
- Steel is one of the world's most recycled materials, because any grade of steel can be recycled into top quality new metal. In North America, two-thirds of all steel is recycled…