Final answer:
The name of the method used to evaluate the given series is a telescoping series. By expanding and simplifying the series, we can see that all the terms cancel out except the first and last term, resulting in a sum of 1.
Step-by-step explanation:
The given series is ∑ₙ=1[infinity]n/n+1.
To find the name of the method used to evaluate this series, we can simplify the expression. Notice that each term in the series is of the form n/(n+1).
We can rewrite this as n/(n+1) = 1 - 1/(n+1).
Now, we can rewrite the series as ∑ₙ=1[infinity](1 - 1/(n+1)).
This is a telescoping series, which means that when we expand and simplify the expression, many terms will cancel out, leaving only a few terms at the beginning and end of the series.
Let's expand the series to see how the cancellation occurs:
∑ₙ=1[infinity](1 - 1/(n+1))
= (1 - 1/2) + (1 - 1/3) + (1 - 1/4) + ...
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ...
Notice that each term cancels out with the next, except the first term of 1 and the last term of -1/(n+1).
Therefore, the series simplifies to 1 + (-1/(n+1)).
To evaluate the series, we take the limit as n approaches infinity:
limₙ→∞(1 + (-1/(n+1)))
= 1 + (limₙ→∞(-1/(n+1)))
= 1 + 0
= 1.