To find the sum of an infinite geometric series, you first need to see if it converges.
This means checking if the common ratio, which we call ( r ), is less than 1 in absolute value. In simpler terms, you want to make sure that ( |r| < 1 ).
If it does converge, you can use this formula to find the sum:
[ S = \frac{a}{1 - r} ]
Here, ( S ) is the sum, ( a ) is the first term, and ( r ) is the common ratio.
Let’s look at an example!
If your series starts with 2, and the common ratio is ( \frac{1}{2} ), you can plug in the numbers:
[ S = \frac{2}{1 - \frac{1}{2}} ]
Solving this gives you:
[ S = 4 ]
It’s really cool how it simplifies like that!
To find the sum of an infinite geometric series, you first need to see if it converges.
This means checking if the common ratio, which we call ( r ), is less than 1 in absolute value. In simpler terms, you want to make sure that ( |r| < 1 ).
If it does converge, you can use this formula to find the sum:
[ S = \frac{a}{1 - r} ]
Here, ( S ) is the sum, ( a ) is the first term, and ( r ) is the common ratio.
Let’s look at an example!
If your series starts with 2, and the common ratio is ( \frac{1}{2} ), you can plug in the numbers:
[ S = \frac{2}{1 - \frac{1}{2}} ]
Solving this gives you:
[ S = 4 ]
It’s really cool how it simplifies like that!