Question

write out the sum.

sum _ { k = 9 } ^ { n } ( - 1 ) ^ { k } ln k

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

Explanation:

Step1: Find the first term

When \(k = 9\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{9}\ln9=-\ln9\)

Step2: Find the second term

When \(k = 10\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{10}\ln10=\ln10\)

Step3: Find the third term

When \(k = 11\), substitute \(k\) into \((-1)^{k}\ln k\).
\((-1)^{11}\ln11=-\ln11\)

Step4: Find the last term

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

Answer:

The first term is \(-\ln9\), the second term is \(\ln10\), the third term is \(-\ln11\), and the last term is \((-1)^{n}\ln n\)