Content
The limiting sum of a geometric series
We have seen that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is
\[S_n = \dfrac{a(1 - r^n)}{1 - r}, \quad\text{for } r\neq 1.\]
In the case when \(r\) has magnitude less than 1, the term \(r^n\) approaches 0 as \(n\) becomes very large. So, in this case, the sequence of partial sums \(S_1,S_2,S_3,\dots\) has a limit:
\[\lim_{n\to \infty} S_n = \lim_{n\to \infty} \dfrac{a(1 - r^n)}{1 - r} = \dfrac{a}{1 - r}.\]
The value of this limit is called the limiting sum of the infinite geometric series. The values of the partial sums \(S_n\) of the series get as close as we like to the limiting sum, provided \(n\) is large enough.
The limiting sum is usually referred to as the sum to infinity of the series and denoted by \(S_\infty\). Thus, for a geometric series with common ratio \(r\) such that \(|r|<1\), we have
\[S_\infty = \lim_{n\to \infty} S_n = \dfrac{a}{1 - r}.\]
Example
Find the limiting sum for the geometric series
- \(1 + \dfrac{1}{3} + \dfrac{1}{9} + \dotsb\)
- \(8 - 6 + \dfrac{9}{2} - \dotsb\).
Solution
- Here \(a = 1\) and \(r = \dfrac{1}{3}\), so the limiting sum exists and is equal to \[ S_\infty = \dfrac{1}{1-\dfrac{1}{3}} = \dfrac{3}{2}. \]
- Here \(a = 8\) and \(r = -\dfrac{3}{4}\), so the limiting sum exists and is equal to \[ S_\infty = \dfrac{8}{1+\dfrac{3}{4}} = \dfrac{32}{7}. \]
Exercise 12
Explain why the geometric series
\[ 1 + \dfrac{1}{1 + \sqrt{2}} + \bigl(3 - 2\sqrt{2}\bigr) + \dotsb \]
has a limiting sum, and find its value.
Exercise 13
By writing the recurring decimal \(0.\overline{12} = 0.121212\dots\) as
\[ \dfrac{12}{10^2} + \dfrac{12}{10^4} + \dotsb, \]
express \(0.\overline{12}\) as a rational number in simplest form.
In general, the limiting sum of an infinite series \(a_1 + a_2 + a_3 + \dotsb\) is the limit, if it exists, of the sequence of partial sums \(S_1, S_2, S_3, \dots\), where
\begin{align*}S_1 &= a_1 \\S_2 &= a_1 + a_2 \\S_3 &= a_1 + a_2 + a_3\end{align*}
and so on. Infinite series are often written in the form
\[\sum_{n=1}^\infty a_n.\]
If the series has a limiting sum \(L\), we say that the infinite series converges to \(L\). This can be written as
\[\sum_{n=1}^\infty a_n = L \qquad\text{or}\qquad a_1 + a_2 + a_3 + \dotsb = L.\]
These two expressions mean that the limit of the sequence of partial sums exists and is equal to the real number \(L\); they can only be used if the infinite series converges.
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FAQs
Sn=a(1−rn)1−r,for r≠1. In the case when r has magnitude less than 1, the term rn approaches 0 as n becomes very large.
What is the limiting value of a geometric sequence? ›
The limiting value of a geometric sequence is the value that the terms of the sequence approach as the number of terms increases. It is also known as the limit of the sequence.
What is the sum of geometric series? ›
The sum of a geometric series Sn, with common ratio r is given by: Sn=n∑i=1ai S n = ∑ i = 1 n a i = a(1−rn1−r) a ( 1 − r n 1 − r ) . We will use polynomial long division formula. Let us see the applications of the geometric sum formulas in the following section.
What is the limiting value of a series? ›
The limit of a series is the value the series' terms are approaching as n → ∞ n\to\infty n→∞. The sum of a series is the value of all the series' terms added together.
What is the formula for the sum of a series? ›
Derivation of Sum of Arithmetic Series Formula
2Sn = n (a1 + an) ⇒ Sn = n(a1 + an )/2. Thus, Sn = n/2(a1 + an). This is one of the formulas to find the sum of arithmetic sequence. Thus, Sn = n/2 [ 2a1 + (n – 1)d], which is another formula to find the sum of arithmetic series.
How to solve a geometric series? ›
Lesson Summary. To review, finite geometric series can be evaluated with the formula a1 ((1 - rn)/(1 - r)) where r is the common ratio and n is the number of terms in the series. Infinite geometric series can be evaluated using a simplified version of this formula, (a1)/(1 - r), but only if r is in between 0 and 1.
How to find the limiting value? ›
To find the limit of a function, use either the direct substitution or factoring method. Direct substitution is best when there is no break, jump, or vertical asymptote at the set value c. It involves substituting the value c for x in the function and simplifying from there.
What is the limit of the geometric mean? ›
The geometric mean of two positive numbers is never greater than the arithmetic mean. So (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ≤ M(x, y) ≤ an.
What is a limiting value in a sequence? ›
A real number L L L is the limit of the sequence x n x_n xn if the numbers in the sequence become closer and closer to L L L and not to any other number. In a general sense, the limit of a sequence is the value that it approaches with arbitrary closeness.
What is the sum of geometric mean? ›
Basically, we multiply the numbers altogether and take the nth root of the multiplied numbers, where n is the total number of data values. For example: for a given set of two numbers such as 3 and 1, the geometric mean is equal to √(3×1) = √3 = 1.732.
Each term of a geometric sequence is formed by multiplying the previous term by a constant number r, starting from the first term a1. Therefore, the rule for the terms of a geometric sequence is an=a1(r)^(n-1).
How do you find the sum of the geometric series if it exists? ›
The general formula for finding the sum of an infinite geometric series is s = a1⁄1-r, where s is the sum, a1 is the first term of the series, and r is the common ratio. To find the common ratio, use the formula: a2⁄a1, where a2 is the second term in the series and a1 is the first term in the series.
What is limiting value? ›
A limit (or limiting value) is the (output) value that a function or sequence approaches as the input approaches some particular value. Limits are an vital part of differentiation and integration and also help us to graph functions. The notation for limits is. limx→cf(x)=l lim x → c f ( x ) = l or f(x)→l as x→c.
Is the limit of a series the sum? ›
The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. If not, we say that the series has no sum.
What is the limiting point of a sequence? ›
Definition. A limit point (or subsequential limit or cluster point) of a sequence {xn} is the limit of any convergent subsequence of {xn}. Recall that the ε-neighborhood of a point a ∈ R is the interval (a − ε, a + ε).
How is a sum of a series related to a limit? ›
The n-th partial sum of a series is the sum of the first n terms. The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. If not, we say that the series has no sum.
What happens if r is greater than 1 in a geometric series? ›
We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you won't get a final answer. The only possible answer would be infinity.
What is geometric limit? ›
Start with a regular triangle of perimeter P. The midlines of the triangle cut it into four equal triangles, each being half as big as the original one. Therefore, the perimeter of each, the middle one in particular, is P/2.
What is the formula for the partial sum of a geometric series? ›
The sum of a finite number of terms of an infinite geometric series is often called a partial sum of the series. Thus, Sn=a+ar+ar2+⋯+arn−1=n−1∑k=0ark. S n = a + a r + a r 2 + ⋯ + a r n − 1 = ∑ k = 0 n − 1 a r k .
