Use the properties of logarithms to simplify the partial sums of the series log( k/k+1 ) Is the series convergent or divergent?
To simplify the partial sums of the series log(k/k+1), we can use the properties of logarithms. Firstly, let's rewrite the series in a different form:
log(k/(k+1)) = log(k) - log(k+1)
Now, let's look at the partial sums:
S_n = log(1) - log(2) + log(2) - log(3) + ... + log(n-1) - log(n)
Notice that the terms log(2), log(3), ..., log(n) cancel out. So, we are left with:
S_n = log(1) - log(n) = -log(n)
Now, let's analyze whether the series is convergent or divergent.
As the terms in the series tend toward infinity (k approaches infinity), the value of log(k) tends to infinity as well. Therefore, we can conclude that the series log(k/k+1) is divergent.