Question

Write an algorithm to find the sum of the following series:

Answer 1

Answer:

The algorithm is as follows:

Input x

sum_a = 0

sum_b = 0

sum_c = 0

for i = 0 to 20

   sum_a = sum_a + x^i

   sum_b = sum_b + (-x)^i

   sum_c = sum_c + x^i/(i+1)

print sum_a, sum_b, sum_c

Explanation:

Required

An algorithm to solve (a), (b) and (c)

For Series (a):

This series is a geometric progression and the common ratio is x

i.e.

$r = x/1 = x^2/x = ...... = x^{n+1}/x^n =x$

So, the sum of the series is:

Sum = Previous Sums + x^i --- where i is between 0 and 20 (inclusive)

For Series (b):

This series is a geometric progression and the common ratio is -x

i.e.

$r = -x/1 = -x^2/x = ...... = -(x^{n+1}/x^n) = -x$

So, the sum of the series is:

Sum = Previous Sums + (-x^i) --- where i is between 0 and 20 (inclusive)

For Series (c):

This series is a neither arithmetic nor geometric progression.

It obeys the following rule:

$\frac{x^i}{1+i}$  --- where i is between 0 and 20 (inclusive)

So, the sum of the series is:

Sum = Previous Sums + $\frac{x^i}{1+i}$