Understanding Infinite Geometric Series in Mathematics
"Explore the concept of infinite geometric series by visualizing the division of a string into halves and learning about convergent series, calculating sums to infinity, and identifying divergent series. Discover how the common ratio affects convergence, with examples and explanations included."
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16 June 2025 The sum of infinite geometric series LO: Identify convergent series and calculate sums to infinity. www.mathssupport.org
Geometric Series Suppose we have a 2 metre length of string . . . 2 m . . . which we cut in half 1 m 1 m We leave one half alone and cut the 2nd in half again 1 2 m 1 2 m 1 m . . . and again cut the last piece in half 1 4 m 1 2 m 1 4 m 1 m www.mathssupport.org
Geometric Series Continuing to cut the end piece in half, we would have in total ? +1 2+1 4+1 8+ ... 1 1 1 m 1 m m m 2 4 8 In theory, we could continue for ever, but the total length would still be 2 metres, so ? +1 2+1 4+1 8+ ... = 2 This is an example of an infinite series. www.mathssupport.org
Geometric Series ? +1 2+1 4+1 The series 8+ ... = 2 r =1 is a Geometric Series with the common ratio 2. . Even though there are an infinite number of terms, this series converges to 2. 2 Sum, Sn 1 4 5 6 2 1 3 Number of terms, n www.mathssupport.org
Sum to infinity We will find a formula for the sum of an infinite number of terms of a Geometric series. This is called the sum to infinity , S ? +1 2+1 4+1 8+ ... = 2 e.g. For the Geometric Series we know that the sum of n terms is given by ? 1 2 u1 = 1 r=1 1 1 (1 rn) (1 r) u1 Sn= ??= 1 1 2 2 ?. As n varies, the only part that changes is This term gets smaller as n gets larger. 1 2 www.mathssupport.org
Sum to infinity ? approaches zero. As n approachesinfinity, 1 2 We write: ? 0 ? ? As n , 0 u1 (1 rn) 1 r u1 So, for r=1 Sn= = 2, 1 r 1 For the series 1 +1 2+1 4+1 = 2 8+ ... ? = 1 1 2 We can write this as ?11 ?? u1 u1 lim = n or S = = 1 r 1 r 1 ? www.mathssupport.org
Divergent series However, if, for example r = 2, rn = 2n n As n increases, also increases. In fact, 2 As n , 2? The geometric series with diverges = = r 2 There is not a sum to infinity www.mathssupport.org
Convergent Series If r is any value greater than 1, the series diverges. Also, if r 1, ( e.g.r = 2), So, again the series diverges. If r = 1, all the terms are the same. If r = 1, the terms have the same magnitude but they alternate in sign. e.g. 2, 2, 2, 2, . . . rn as n A Geometric Series converges only if the common ratio r lies between 1 and 1. ?1 for 1 < r < 1 ? = 1 ? This can also be written as | r | < 1 www.mathssupport.org
Convergent Series Example 1 For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity. 18 + 6 + 2 + ., Solution: find the sum to infinity ?1 ? = 1 ? 18 u1 = 18 r = =1 6 18 ? = 1 3 3 1 so r does satisfy 1 < r < 1 the series converges. S = 27 The series converges to 27 www.mathssupport.org
Convergent Series Example 2 For the following geometric series, write down the value of the common ratio, r, and decide if the series converges. If so, find the sum to infinity. 2 8 2 Solution: u1 = 2 1 2 2 4 1 1 1 + + + + . . . 32 ?1 ? = find the sum to infinity 1 ? = 1 2 r = ? = 1 1 so r does satisfy 1 < r < 1 the series converges. 4 S =8 5=1.6 The series converges to 1.6 www.mathssupport.org
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