If 1+(1/2^2)+(1/3^2)+(1/4^2)+(1/5^2)+...= x then the value of 1+(1/2^2)+(1/4^2)+(1/6^2)+(1/8^2)+... is?
Let's write out the terms so it's a little more readable.
1 + 1/4 + 1/9 + 1/16 + 1/25 + ... = x
and
1 + 1/4 + 1/16 + 1/36 + 1/64 +... = ?
Can you see that the second series is
1 + 1/4(1 + 1/4 + 1/9 + 1/16 + ...)=1+1/4x ?
To find the value of the second series, we can use the formula for the sum of an infinite geometric series.
The formula for the sum of an infinite geometric series is:
S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio between consecutive terms.
In the second series, the first term a is 1, and the common ratio r is 1/4.
Plugging these values into the formula, we get:
S = 1 / (1 - 1/4) = 1 / (3/4) = 4/3 = 1.33 (rounded to two decimal places)
Therefore, the value of the second series 1 + 1/4 + 1/16 + 1/36 + 1/64 + ... is approximately 1.33.