Question

write out the sum.

sum _ { k = 2 } ^ { n } ( - 1 ) ^ { k } ln k

find the first, second, third, and last terms of the sum.

sum _ { k = 2 } ^ { n } ( - 1 ) ^ { k } ln k = square + square + square + cdots + square
(type an exact answer in simplified form.)

Explanation:

Step1: Find the first term

When \(k = 2\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{2}\ln 2=\ln 2\)

Step2: Find the second term

When \(k = 3\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{3}\ln 3=-\ln 3\)

Step3: Find the third term

When \(k = 4\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{4}\ln 4=\ln 4\)

Step4: Find the last term

When \(k = n\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{n}\ln n\)

Answer:

\(\ln 2-\ln 3+\ln 4+\cdots+(-1)^{n}\ln n\)