Let a_n>0 for all n and suppose \sum a_n < \infty. Let b_n=1-1/n. Show that \sum a_n ^{b_n} <\infty.
Hint: split the sum into two parts, one in which a_n ^{b_n} <a_n and one where a_n ^{b_n} is dominated by a convergent geometric series.
Let a_n>0 for all n and suppose \sum a_n < \infty. Let b_n=1-1/n. Show that \sum a_n ^{b_n} <\infty.
Hint: split the sum into two parts, one in which a_n ^{b_n} <a_n and one where a_n ^{b_n} is dominated by a convergent geometric series.