Calculate some of each of the following series
2 + 1/3 + 4 + 1/4 + 8 + 1/8 + 16 + 1/16+....(to terms)
The given series can be written as:
2 + 4 + 8 + 16 + ... + 2^n + 1/n + ...
To find the sum of the series up to n terms, we can use the formula for the sum of a geometric series:
S = a(1 - r^n) / (1 - r)
where a is the first term (in this case, 2) and r is the common ratio (in this case, 2).
For the first part of the series (2 + 4 + 8 + 16 + ... + 2^n), the sum can be calculated as:
S1 = 2(1 - 2^n) / (1 - 2)
= 2 - 2^(n+1)
For the second part of the series (1/n), we need to calculate the sum of 1/3 + 1/4 + 1/8 + 1/16 + ... + 1/2^n. This is a geometric series with a common ratio of 1/2.
Using the formula again, we get:
S2 = (1/3)(1 - (1/2)^n) / (1 - (1/2))
= (1/3)(1 - (1/2)^n) / (1/2)
= (2/3)(1 - (1/2)^n)
Adding both parts together, we get:
S = S1 + S2
= 2 - 2^(n+1) + (2/3)(1 - (1/2)^n)
So, the sum of the given series up to n terms is 2 - 2^(n+1) + (2/3)(1 - (1/2)^n).