Test 2: Real Analysis
Instruction: Create a pdf file with all the solutions, upload file on Google Drive and share the link in the form shared at the bottom of the page
Total Marks: 75
1) Define the following terms with an example of each:
i) Limit
ii) Continuity
iii) Differentiability
iv) Sequence
v) Series
[10M]
2) Define below terms with an example of each:
i) Limit point
ii) Derived Set
iii) Uniform Continuity
iv) Pointwise Convergence
v) Uniform Convergence
[10M]
3) In the context of Real Analysis, write 1 sentence each including the below statements:
i) “necessary and sufficient”
ii) “necessary but not sufficient”
iii) “sufficient but not necessary”
[6M]
4) Write the statements of following tests:
i) P-Series Test
ii) Root Test
iii) Raabe’s Test
iv) Leibnitz test
vi) Gauss’s Test
[10M]
5) Write the statements of following tests in the context of convergence of series of functions:
i) Weierstrass M-Test
ii) Abel Test
iii) Dirichlet Test
[6M]
6) With clear explanations, give one example each of below:
i) A convergent sequence with the corresponding series diverging
ii) A convergent sequence with the corresponding series converging
iii) A series that converges pointwise but not uniformly
iv) A series that converges conditionally
[8M]
7) Show that \(\int_{0}^{\infty} \dfrac{x d x}{1+x^{4} \sin ^{2} x}\) is divergent.
[12.5M]
8) Show that \(\int_{0}^{\infty} e^{-\alpha x} \dfrac{\sin x}{x} d x, a \geq 0\) is convergent.
[12.5M]