Find the sum of the following series.
14/n-=1 (2n+1)
I don't see a series. I do see a -= which I cannot decode.
And what does 1 (2n+1) mean?
Use ^ for exponents, as in 2^n or (-1)^(n-1)
Use parentheses if needed
1/n+2 is not the same as 1/(n+2)
To find the sum of the given series, we need to evaluate the sum of each term in the series and then add them together.
The given series is:
14/n = 1 / (2n + 1)
First, let's try to find the sum of each term in the series. We can do this by considering the values of n.
When n = 1:
14/1 = 1 / (2(1) + 1) = 14/1 = 14
When n = 2:
14/2 = 1 / (2(2) + 1) = 14/5
When n = 3:
14/3 = 1 / (2(3) + 1) = 14/7
...and so on.
We notice that the value of the denominator in each term is (2n + 1).
To find the sum of each term, we can write an equation:
S = (14/1) + (14/2) + (14/3) + ...
We can rewrite each term in the series as:
S = 14/1 + 14/2 + 14/3 + ... + 14/n
Now, let's simplify the expression. We can factor out the common factor 14:
S = 14 * (1/1 + 1/2 + 1/3 + ... + 1/n)
The expression (1/1 + 1/2 + 1/3 + ... + 1/n) represents the harmonic series, which does not have a simple closed form. It is known as the harmonic number H(n).
Therefore, the sum of the given series is:
S = 14 * H(n)
To find the value of H(n), you can use a calculator or a computer program specifically designed to calculate the harmonic numbers.