Calculus 2222222222@2@22222222222222).pdf
- 1. 7.2 Geometric Series At the end of the lessons students are able to recognise a geometric series determine whether a geometric series convergent or divergent find its limiting sum
- 2. Definition of series A series is the sum of the terms of a sequence. We write the sum of the first n terms of a sequence as , where General Form
- 3. is called the first partial sum. is called the second partial sum and so on. are called sequences of partial sums. Note: is a finite series is an infinite series i) ii)
- 4. A geometric series can be written as a + a r + a r2 + a r3 + . . .+ a r n – 1 + . . . First Term Common Ratio Second Term Third Term Fourth Term nth Term a r0 = a r0 = 1 where r ≠ 0 Tn = a r n - 1 General term
- 5. By looking at the pattern, identify common ratio, r which is in the form of rn S ∞ = n = 1 ∞ a r n - 1 = a 1 - r i) l r l < 1, So, the series converges the series diverges ii) l r l 1, The common ratio, r is given by r = = = . . . = T 2 T 1 T 3 T 2 T n T n - 1
- 6. Solution
- 7. Determine whether the series convergent or divergent. If it convergent, find the infinite sum.
- 8. Solution a) Given, So, the series diverges b) Given,
- 9. So, series converges So, series converges
- 10. So, the series diverges
- 11. Exercise Determine whether the series convergent or divergent. If it convergent, find the If it convergent, find the infinite sum.