Geometric Series

Definition and general term of geometric series, finite and infinite series, convergence

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Exercises

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Key Skills

Finite Geometric Series

SL Core 1.3

The sum of the first n terms in a geometric sequence is given by:

Sn=r−1u1(rn−1)=1−ru1(1−rn)📖

Infinite Geometric Series

SL AA 1.8

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:

S∞=1−ru1,∣r∣<1📖

Convergence

SL AA 1.8

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

u4=u1r3=8⋅r3=2k+1⇒r3=82k+1

Now if −1<r<1, then −1<r3<1:

−1<82k+1<1

−8<2k+1<8

−29<k<27

Finite Geometric Series

SL Core 1.3

The sum of the first n terms in a geometric sequence is given by:

Sn=r−1u1(rn−1)=1−ru1(1−rn)📖

Infinite Geometric Series

SL AA 1.8

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:

S∞=1−ru1,∣r∣<1📖

Convergence

SL AA 1.8

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

u4=u1r3=8⋅r3=2k+1⇒r3=82k+1

Now if −1<r<1, then −1<r3<1:

−1<82k+1<1

−8<2k+1<8

−29<k<27