1 · IB Maths AA
Arithmetic and geometric series
Set the first term and the common difference or ratio, then watch the terms (bars) and partial sums Sₙ. For a geometric series with |r| < 1 the partial sums close in on the sum to infinity.
Try this
- Geometric, u₁ = 8, r = 0.5: S∞ = 16. Increase n until S∞ − Sₙ is below 0.01. How many terms are needed?
- Keep u₁ = 8 but set r = −0.5. Describe how the partial sums approach S∞ compared with r = 0.5.
- Arithmetic, u₁ = 20, d = −2: find the value of n for which Sₙ returns to 0, then check it with the Sₙ formula.
Common error: using S∞ = u₁/(1 − r) without first checking that |r| < 1; when |r| ≥ 1 the partial sums have no limit, so there is no sum to infinity.