List of top Mathematics Questions on Principle of Mathematical Induction asked in CUET (PG)

If f(z)=1z2−3z+2f(z)=\frac{1}{z^2-3z+2}is expanded in the region |z|<1, then
The given series x1.3+x22.4+x33.5+......,(x>0)\frac{x}{1.3}+\frac{x^2}{2.4}+\frac{x^3}{3.5}+......,(x\gt0)is convergent in the interval
The infinite series ∑n=1∞(1+1n)−n2\displaystyle\sum_{n=1}^{∞} (1+\frac{1}{n})^{-n^2} is:
The infinite series ∑n=1∞3n4n+2\displaystyle\sum_{n=1}^{∞} \frac{3^n}{4^{n+2}} is:
Match List I with List II
LIST I LIST II
A.Series ∑n=1∞1n32\displaystyle\sum_{n=1}^{∞} \frac{1}{n^\frac{3}{2}} isI.Monotone and 
convergent both
B.Series ∑n=1∞3nn2\displaystyle\sum_{n=1}^{∞} \frac{3^n}{n^2} isII.e−2e^{-2}
C.lim⁡n→∞(n+1n+2)2n+1\lim\limits_{n \to \infty} (\frac{n+1}{n+2})^{2n+1}III.Divergent to ∞
D.sequence xn=1+12!+13!+…1n!x_n=1+\frac{1}{2!}+\frac{1}{3!}+…\frac{1}{n!} for n∈NIV.Convergent
Choose the correct answer from the options given below:
The value of lim⁡n→∞1n[1+212+313+...n1n]\lim\limits_{n \to \infty} \frac{1}{n} [1+2^{\frac{1}{2}}+3^{\frac{1}{3}}+...n^{\frac{1}{n}}] is