Find the sum of the series 1-1/2+1/4-1/8+...to n terms.
same old same old, with
a = 1
r = -1/2
the arithmetic mean of two numbers exceeds the geometric mean by 3/2 and geometric mean exceeds the harmonic by 6/5. find the numbers.
To find the sum of the given series, let's start by examining the pattern of the terms. The series alternates between positive and negative terms with each term being half of the previous term.
We can rewrite the terms of the series as follows:
1 - 1/2 + 1/4 - 1/8 + ...
To determine the sum of the series, we can use the formula for the sum of a geometric series. The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
where:
S = sum of the series
a = first term of the series
r = common ratio
In our series, the first term is 1 and the common ratio is -1/2.
Now, since we have an infinite series, we need to determine whether it converges or diverges. To do this, we need to find the absolute value of the common ratio (|-1/2| = 1/2) and check if it is less than 1. In our case, the absolute value of the common ratio is less than 1, so the series converges.
Therefore, we can apply the formula to find the sum of the series:
S = a / (1 - r)
S = 1 / (1 - (-1/2))
S = 1 / (1 + 1/2)
S = 1 / (3/2)
S = 2/3
Hence, the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... is 2/3.