Use the sequence of partial sums to determine the sum of the series (1/sqrtk-1/sqrtk+1). What’s the sum?
To determine the sum of the series (1/sqrt(k) - 1/sqrt(k+1)), we can use the sequence of partial sums.
First, let's write out some terms of the sequence to observe any patterns:
S1 = (1/sqrt(1)) - (1/sqrt(2))
S2 = (1/sqrt(1)) - (1/sqrt(2)) + (1/sqrt(2)) - (1/sqrt(3))
S3 = (1/sqrt(1)) - (1/sqrt(2)) + (1/sqrt(2)) - (1/sqrt(3)) + (1/sqrt(3)) - (1/sqrt(4))
...
Notice that many terms cancel each other out. For example, in S2, (1/sqrt(2)) - (1/sqrt(2)) = 0. Similarly, in S3, both (1/sqrt(2)) and (1/sqrt(3)) are present, but since they have opposite signs, they cancel each other out.
Simplifying the sequence, we can now write it as:
S1 = (1/sqrt(1)) - (1/sqrt(2))
S2 = (1/sqrt(1))
S3 = (1/sqrt(1))
...
We can see that all terms after S1 are the same. Therefore, we can conclude that the sum of the series is S1.
Now, let's calculate S1:
S1 = (1/sqrt(1)) - (1/sqrt(2))
= 1 - 0.7071 (approximating the square root of 2 as 1.4142)
≈ 0.2929
Therefore, the sum of the series (1/sqrt(k) - 1/sqrt(k+1)) is approximately 0.2929.