Answer Happy

T/F?
1. The series n=0∑∞​an​xn and n=1∑∞​nan​xn−1 have the same radius of convergence. 2. The series n=0∑∞​an​xn and n=1∑∞​nan​xn−1 have the same interval of convergence. 3. Suppose that n=0∑∞​an​xn has radius of convergence R>0. Let f(x)=n=0∑∞​an​xn. Then ∫f(x)dx=n=0∑∞​(∫an​xndx)=C+n=0∑∞​an​n+1xn+1​ 4. Suppose that n=0∑∞​an​xn has radius of convergence R>0. Let f(x)=n=0∑∞​an​xn. Then ∫ab​f(x)dx=n=0∑∞​an​n+1bn+1​−n=0∑∞​an​n+1an+1​