Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Sequences & Series (Convergence)
Let {bn} and {cn} be sequences of real numbers. Then a necessary and sufficient condition for the sequence of polynomials fn(x) = bnx + cnx2 to converge uniformly to 0 on the real line is
1
\(\displaystyle\lim _{n \rightarrow \infty} b_{n}=0\) and \(\displaystyle \lim _{n \rightarrow \infty} c_{n}=0\)
2
\(\displaystyle \sum_{n=1}^{\infty}\left|b_{n}\right|<\infty\) and \(\displaystyle \sum_{n=1}^{\infty}\left|c_{n}\right|<\infty\)
3
There exists a positive integer N such that bn = 0 and cn = 0 for all n > N
4
\(\displaystyle \lim _{n \rightarrow \infty} c_{n}=0\)