If each fn is continuous on [a, b] and (sum of)fn converges uniformly on [a, b] then (sum of)Mn
converges, where Mn = max|fn(x)|for x belongs to [a,b].
Is it true?
Thx in advance
.
No, it's not true. But to find a counterexample you have to use some test other than the M-test to prove the uniform convergence. That probably means using the uniform version of the Dirichlet or Abel tests for convergence. A typical counterexample is the series $\displaystyle \sum\frac{(-x)^n}n$ on the interval [0,1]. This converges uniformly by the uniform Dirichlet test. But $\displaystyle \max\{|(-x)^n/n|:x\in[0,1]\} = 1/n$ and $\displaystyle \sum 1/n$ does not converge.