Geometric Series: Find Sum of Infinity - 9-32-n

In summary, the formula for finding the sum of an infinite geometric series is S_{\infty} = \frac{a}{1-r}, where "a" is the first term and "r" is the common ratio. If the common ratio is less than 1, the sum of the infinite series exists and is equal to a/(1-r). However, if the common ratio is greater than or equal to 1, the sum does not exist.
  • #1
ChelseaL
22
0
Given that the sum of the first n terms of series, s, is 9-32-n

Find the sum of infinity of s.

Do I use the formula S\infty = \frac{a}{1-r}?
 
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  • #2
ChelseaL said:
Given that the sum of the first n terms of series, s, is 9-32-n

Find the sum of infinity of s.

Do I use the formula S\infty = \frac{a}{1-r}?

I can't tell what your geometric terms are. Are they supposed to be $9 - 3^{2-n}\;for\;n\in \mathbb{N}$? Are you SURE that's a geometric sequence?
 
  • #3
I'm not exactly sure what the topic is, but this is the part before it.
https://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/geometric-series-24025.html
 
  • #4
What you can do is:

\(\displaystyle S_{\infty}=\lim_{n\to\infty}S_n\)
 
  • #5
How do I work it out from there?
 
  • #6
ChelseaL said:
How do I work it out from there?

Have you worked with limits before?
 
  • #7
This is my first time hearing this term. I don't think my lecturer expects us to use that method since we never covered it in class.
 
  • #8
ChelseaL said:
This is my first time hearing this term. I don't think my lecturer expects us to use that method since we never covered it in class.

Okay, then the formula you first cited works in this case:

\(\displaystyle S_{\infty}=\sum_{k=0}^{\infty}a_n=\frac{a}{1-r}\)

It is my understanding your sum is:

\(\displaystyle \sum_{k=1}^{\infty}a_n\)

In this case, you would need to write:

\(\displaystyle S_{\infty}=\frac{a}{1-r}-a=\frac{ar}{1-r}\)
 
  • #9
Consider the finite geometric series \(\displaystyle S= a+ at+ at^2+ at^3+ \cdot\cdot\cdot+ at^n\). Subtract a from both sides: \(\displaystyle S- a=at+ at^2+ at^3+ \cdot\cdot\cdot+ at^n\). Factor "t" out of the right side: \(\displaystyle S- a= t(a+ at+ at^2+ \cdot\cdot\cdot+ at^{n- 1})\). The sum on the right is almost the same as the original sum- it is only missing the "\(\displaystyle at^n\)" term. Add and subtract \(\displaystyle at^n\) inside the parentheses:
\(\displaystyle S- a= t(a+ at+ at^2+ \cdot\cdot\cdot+ at^{n- 1}+ at^n- at^n)= t(a+ at+ at^2+ \cdot\cdot\cdot+ at^{n- 1}+ at^n)- at^{n+1}= tS- at^{n+1}\)

Add \(\displaystyle at^{n+1}\) to both sides:
\(\displaystyle S- a+ at^{n+ 1}= tS\)
\(\displaystyle S- tS= (1- t)S= a- at^{n+1}= a(1- t^{n+1})\)
Finally, divide both sides by 1- t:
\(\displaystyle S= \frac{a(1- t^{n+1})}{1- t}\)

The sum of the infinite series, \(\displaystyle \sum_{n=0}^\infty at^n\) is the limit of that as n goes to infinity. If \(\displaystyle t\ge 1\), that limit, so that sum, does not exist. If \(\displaystyle t<1\), \(\displaystyle t^{n+ 1}\) goes to 0 so
\(\displaystyle \sum_{n=0}^\infty at^n= \frac{a}{1- t}\).
 

Related to Geometric Series: Find Sum of Infinity - 9-32-n

1. What is a geometric series?

A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.

2. How do you find the sum of a geometric series?

The sum of a finite geometric series can be found using the formula Sn = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. To find the sum of an infinite geometric series, we use the formula S = a / (1 - r), as long as the absolute value of r is less than 1.

3. What is the common ratio in a geometric series?

The common ratio in a geometric series is the number that is multiplied by each term to get the next term. It is represented by the letter r and can be found by dividing any term by the previous term.

4. What does it mean to find the sum of infinity in a geometric series?

Finding the sum of infinity in a geometric series means finding the sum of an infinite number of terms in the series. This is only possible when the common ratio is between -1 and 1, as the sum will approach a finite value as the number of terms increases.

5. How can we use geometric series to solve real-world problems?

Geometric series can be used to model real-world situations where there is a constant growth or decay. For example, population growth, compound interest, and radioactive decay can all be represented by geometric series. By finding the sum of these series, we can make predictions and solve problems related to these scenarios.

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