Practice problems -3(PG Mathematics)

1. Find lim $\sum_{k=m}^{\infty}a_k$ as m tends to
$\infty$ , if the series $\sum_{k=1}^{\infty}a_k$ is convergent.

2. Every k- cell is _____

3. Give an example of a non analytic function for which CR equations are satisfied.

4. Prove that if u and v are harmonic functions need not to imply f=u+iv is analytic.

5. Verify the analyticity of the functions $|z|^2$ and $z \overline{z}$.

6. The Petersen graph is __________ (Eulerian/ Hamiltonian / Both/ Neither)

7. The radius of curvature at any point on the Helix $x=acos \theta$, $y=asin \theta$ and $z=a \theta tan \alpha$ is __________

8. Any countable set in real line has measure _____.

9. The function $f(z)=|z|^2$, is ____ (Analyic everywhere/Analytic nowhere/ Analytic at z=0 /Noneofthese)

10. Any __________ separable space is separable (Subspace/ Open subspace / Closed subspace).