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Arithmetico Geometric Series

A series is said to be in arithmetico geometric series if each of its term is the product of the corresponding terms of an AP and a GP.
For example, 1, 2x, 3x2, 4x3,…
In the above series, the first part of this series is in an AP (1, 2, 3, 4,…) and the 2nd part is (x0x1x2x3,….) in a GP.
Sum of n terms of any arithmetico geometric series (AGS)
The sum of n terms of any AGS a, (a + d)r, (a + 2d)r2,…is given by
Description: 2724.png. rn if r  1.
Sn=Description: 2733.png[2a + (n  1)d], if r = 1
Sum of infinite terms of any arithmetico geometric series (AGS)
Description: 2737.png 

However, I would suggest students to desist from using these formulae. They should use the standard process to find out the sum of any AGS which is given below:
Let N be the sum of the arithmetico geometric series. Then each term of the series is multiplied by r (the common ratio of GP) and is written by shifting each term one step rightward, and then by subtracting rN from N to get (1r) N. Thus N is finally obtained.
What is the sum of the following series till infinity: 1 + 2x + 3x2 + 4x3 + …, |x| <1
Assume S = 1 + 2x + 3x2 + 4x3 +…(1)
Multiplying S by xx. S = x + 2x2 + 3x3 + 4x4 + …(2)
Subtracting (2) from (1)
S – x S = 1 + (x + x2 + x3 +…∝)
S(1 − x) = 1 + (x + x2 + x3 +…∝) = Description: 2746.png
Description: 2755.png

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