Finite geometric series with sum Sn=1−ru1(1−rn),
The sum of the first n terms in a geometric sequence is given by:
If a geometric sequence has a common ratio ∣r∣<1, then each term will be smaller than the previous term. As the terms get smaller and smaller, the sum of all the terms approaches a finite value:
A geometric series is said to converge if S∞ is finite - which means ∣r∣<1⇔−1<r<1.
Example
A geometric sequence has u1=8 and u4=2k+1. For what value(s) of k does the corresponding geometric series converge?
We have
Now if −1<r<1, then −1<r3<1:
Nice work completing Geometric Series, here's a quick recap of what we covered:
Exercises checked off
Finite geometric series with sum Sn=1−ru1(1−rn),
The sum of the first n terms in a geometric sequence is given by:
If a geometric sequence has a common ratio ∣r∣<1, then each term will be smaller than the previous term. As the terms get smaller and smaller, the sum of all the terms approaches a finite value:
A geometric series is said to converge if S∞ is finite - which means ∣r∣<1⇔−1<r<1.
Example
A geometric sequence has u1=8 and u4=2k+1. For what value(s) of k does the corresponding geometric series converge?
We have
Now if −1<r<1, then −1<r3<1:
Nice work completing Geometric Series, here's a quick recap of what we covered:
Exercises checked off