1. Series

2. For the arithmetic series a + (a + d) + (a + 2d) +…
the sum of n terms is given by
Sn = [ 2a + (n – 1) d ] Or
Sn = (t1 + tn)

3. Find S22 for the AP, 3, 8, 13, …
S22 = [ 2a + ( 22 – 1) d ]
For this AP, a = 3 and d = 5
S22 = [ 2 x x 5 ]
= 11 [ ]
= 11 x 111
S22 = 1221

4. Find the sum of the AP 7, 11, 15, …, 203.
Here a = 7 and d = 4, but in order to apply the formula for summing the series, we need to know n.
tn = a + (n - 1) d
203 = (n – 1) 4
196 = 4 (n – 1)
49 = n – (dividing by 4)
50 = n
So 203 is the 50th term
Now find S50. Use Sn = (t1 + tn)
S50 = ( )
S50 = 25 x 210
S50 = 5250

5. Find S500 for the AP 7, 11, 15, …
Find S100 for the AP 95, 92, 89, …
Find S50 for the AP -36, -31, -26, …
Find S20 for the AP -18, -20, -22, …
(a) Find the number of terms in this AP 5, 8, 11, …92.
(b) Find the sum of the terms of the AP in (a).
6. Find the sum of the AP 2, 7, 12, … 87.

6. For the geometric series a + ar2 + ar3 +…
the sum of n terms is given by or
Use this when
r > 1
Use this when
r < 1

7. Find S20 for the GP 3, 15, 75, 225,… Here a = 3, r = 5 and n = 20
S20 = 7.15 x 1013

8. Find S50 for the GP: 2, 6, 18, 54,… Here a = 2 and r = 3, n = 50
= 350 – 1 (leave in this form)

9. Try these. Leave large powers in index form
Find S20 for the GP: 1, 2, 4, 8, …
Find S50 for the GP: 2, -6, 18, -54, …
Find S12 for the GP: 1, 1/3, 1/9, 1/27, …
Find S30 for the GP: 4, -2, 1, - ½ , ¼,…
Find S15 for the GP: 2, 0.2, 0.02, 0.002, …

10. Infinite series – the limit of a sum
A series may have a limit
For some series, as more terms are added on, the sums produced approach nearer and nearer to a particular value. The value ℓ is called the SUM TO INFINITY of the series and we say the series is CONVERGENT.
MX233 pg33

11. Infinite series – the limit of a sum
If, as more terms are added Sn gets larger or more negative without approaching a particular value, or oscillates, then the series does not have a limit and we say it is DIVERGENT.

12. Are these series convergent or divergent?…
S1 = 2 S2 = 8 S3 = 18
S4 = 32 S5 = 50 S6 = 72
S7 = 98 S8 = 128 S9 = 162
As n increases Sn is also increasing without approaching a particular value so the series is divergent.

13. Are these series convergent or divergent?
/3 + 1/9 + 1/27 + …
S1 = 9 S2 = 12 S3 = 13
S4 = S5 = S6 =
S7 = S8 = S9 =
The sums appear to be getting closer to So we say that the series is convergent and the sum to infinity is 13.5.
= 13.5

14. For each of the following series: Find S1 to S10 at least.
State whether the series is convergent or divergent and a value accordingly. …
1 + 1/3 + 1/9 + 1/27 + … …
¼ + ½ …
MX233 page 35

15. Limit of an arithmetic series
Any arithmetic series is divergent

16. Limit of a geometric series
Not all geometric series have a limit
If r ≥ 1, Sn increases without end
If r ≤ -1, Sn oscillates.
If -1 < r < 1 Sn will have a limit and the series is convergent

17. Limit of a geometric series
Formula for the sum to infinity of a series is:
For -1 < r < 1. BUT as n takes larger and larger values, rn gets closer and closer to zero.
Therefore the formula becomes

18. Limit of a geometric series
For the series ¼ + 1/16 + …
a = 4, r = ¼
aaaaa
aaaaa

19. Which of the following series are convergent
Which of the following series are convergent? For those which are, find the value of S∞.
Is a geometric series
9 + 9/2 + 9/4 + 9/8 + …
With a = 9 and 2 = 1/2 . Since -1 < r < 1, the series is convergent. a a

20. Which of the following series are convergent
Which of the following series are convergent? For those which are, find the value of S∞. …
Is an arithmetic series and is therefore divergent
3. 1/ …
Is a geometric series with a = 1/10 and r = 10. since r > 1, the series is divergent