Solve your IGNOU Doubts

Question:

Show that the series $\sum_{n=1}^{\infty }\frac{sin\, n\theta }{n}$ does not converge uniformly on the interval $\left ]0,2\pi \right [$ $.$

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Question:

Check whether the series $f(x)=2x+7\sum_{n=1}^{\infty }\frac{n^{2}x^{5}}{n^{4}+x^{3},},x\in[0,\alpha ]$ is uniformly convergent or not, wher $\alpha \in \mathbb{R}^{+}.$

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Question:

Show that the function $f\, :\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=2x+7$ has an inverse by applying the inverse function theorem. Find its inverse also.

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Question:

Use the Fundamental Theorem of Integral Calculus to evaluate the integral

$\int_{0}^{1}\left ( 2x\, sin\frac{1}{x}-cos\frac{1}{x} \right )dx.$

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Question:

Using Weiestrass M-test, show that the following series converges uniformly. $\sum_{n=1}^{\infty }n^{3}\: x^{n}\: ,x\in \left [ -\frac{1}{3},\frac{1}{3} \right ].$

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Question:

Use Cauchy’s Mean Value Theorem to prove that:

$\frac{cos\, \alpha -cos\, \beta }{sin\, \alpha-sin\, \beta }=tan\theta ,0< \alpha < \theta< \beta < \frac{\pi }{2}$

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Question:

Show that $\sum_{n=1}^{\infty }(-1)^{n+1}\frac{5}{7n+2}$ is conditionally convergent.

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Question:

Test the following series for convergence.

(i) $\sum_{n=1}^{\infty }n\: x^{n-1}\, ,x> 0\: .$

(ii) $\sum_{n=1}^{\infty }\left [ \sqrt{n^{4}+9}-\sqrt{n^{4}-9} \right ]$

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Question:

Suppose that $f\: :[0,2]\rightarrow \mathbb{R}$ is continuous on $[0,2]$ and differentiable on $]0,2[$ and that f )0( = ,0 f )1( = ,1 f )2( = .1 (i)

Show that there exists $c_{1}\in (0,1)$ such that ${f}'(c_{1})=1.$

Show that there exists $c_{2}\in (0,1)$ such that ${f}'(c^{2})=0,$

(iii) Show that there exists $c\in (0,2)$ such that . ${f}'(c)=\frac{1}{3}.$

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Question:

Let $f$ be a differentiable function on $[\alpha ,\beta ]$ and $x\in [\alpha ,\beta ].$ Show that, if $f{}'(x)=0$ and $f{}'(x)>0,$ then $f$ must have a local maximum at $x$ .

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Question:

Let $f:[01]\rightarrow$ $\mathbb{R}$ be a function defined by $f(x)=x^{m}(1-x)^{n}\: ,$ where $m,n\in \mathbb{N}.$ Find the values of m and n such that the Rolle’s Theorem holds for the function $f.$

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Question:

Determine the local minimum and local maximum values of the function f defined by $f(x)=3-5x^{3}+5x^{4}-x^{5}\: .$

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Question:

Prove that a strictly decreasing function is always one-one

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Question:

Find the following limit

$\lim_{x\rightarrow 0}\frac{1-cos\: x^{2}}{x^{2}sin\: x^{2}}$

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Question:

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

$\left\{\begin{matrix} -x^{2}\: , &when\: x\leq 0 \\4-5x, &when\: 0< x\leq 1 \\3x-4x^{2}\: , &when\: 1< x\leq 2 \\ -12x+2x\: , & when\: x< 2 \end{matrix}\right.$

Also check whether the function f is derivable at x = .1

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Question:

Evaluate

$\lim_{n\rightarrow \infty }$ $\,$ $\left [ \frac{n}{1+n^{2}}+\frac{n}{4+n^{2}}+\frac{n}{9+n^{2}}+\cdots+\frac{n}{2n^{2}} \right ].$

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Question:

d) Show that $\left ( \frac{1}{n^{2}+n+1} \right )_{n\in \mathbb{N}}$ is a Cauchy sequence.

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Question:

Let $(a_{n})_{n\in \, \mathbb{N}}$ be any sequence. Show that $\lim_{n\rightarrow \infty }a_{n}=L$ iff for every ε > 0, there exists some $N\in \! \mathbb{N}$ such that n ≥ N implies $a_{n}\in N_{\varepsilon }(L).$