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Show the series Σ x/(1+n^2.x^2 is uniformly convergenr in [ᾰ,1], for any ᾰ>1.
Show that the series Σ x/(1+n^2.x^2) is uniformly convergent using Weierstrass m test
Suppose that the sequence (sn) converges to s and sn ≤ A for every n. Show that s≤A
Let f: [0,1] to R ne a function defined by
f(x)= 1-x^2
Let P1= { 0,1/2,2/3,1}
P2= { 0,1/4,1/2,3/4,1}
be two partition of the interval [0,1]. Calculate L(P2,f) and U(,P1,f)
Show that the notation {Xi} i€I implicitly involves the notion of function.
Using the definition prove the convergence of the following sequences:
(i) lim Cos nα
n+1
= 0 (ii) lim n
3−1
3+2n3 =
1
2
(iii) lim 1
(n+1)
2 +1
=0
Further, find all x∈ ℝ that satisfy the inequality: |x|+ |x + 1|<2
”A real number is rational if and only if it has a periodic decimal expansion.” Define the present usage of the word periodic and prove the statement.
2. Using the definition prove the convergence of the following sequences:
(i) lim Cos nα
n+1
= 0 (ii) lim n
3−1
3+2n3 =
1
2
(iii) lim 1
(n+1)
2 +1
=0
Further, find all x∈ ℝ that satisfy the inequality: |x|+ |x + 1|<2
Check the convergence or divergence of the following series
(i) ∑ √𝑛
3 + 1 − √𝑛
3 − 1
∞
𝑛=1
(ii) ∑
7
𝑛+1
9
𝑛
∞
𝑛=1
(iii) ∑
𝑛
3
3
𝑛
∞
𝑛=1
(iv) 1
𝑒
+
4
𝑒
2 +
27
𝑒
3 …
Evaluate,
lim(√n/√n^2+ √n/√(n+3)^2+...√n/√(7n- 3)^2
n→∞
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