1. Apply Parseval's theorem to Q(t)= -{Q(+27) and use it to evaluate a sum rule for Σ1 k=1 |t|

Q: For positive integers n ≥ 1, consider the functions 2 A (a) Show that (b) Show that Sn(x) = ㅠ [ Sn =…

A: Given: snx=2πsinnx. To show,: (a) ∫0πsnx2dx=1 for all n. (b) ∫0πsnxsmxdx=1 for all n≠m.

Q: p(x) = a + ax + a,x² + ... + 10x?

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Q: Prove = cos for some x in (0, π/2).

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Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

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Q: (a) Prove that Σ³ = n²(n+1) ². j=1 (b) Evaluate fr³ de directly from the definition, without using…

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Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any r e [0, 1] but | f(x)dx = 0.…

A: Let us consider a function f:0,1→ℝ such that fx≥0 for all  x∈0,1.  Suppose fxn be a sequence of…

Q: 11.Solve the polynomial inequality (4 – x?) (x² – 1) 2 0 [T: 4]

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Q: (3) Consider the function defined as : R -> R>0 if laT 1 2 p(x) 0 if a> np(nx), and let Kn(x) be the…

A: Consider the given function which is defined as,

Q: 5. (a) Let f: R R be defined by f(x) = x², let g: RR be defined by g(x) = sin x, and let h: R→ R be…

A: Step 15(a) The given functions are:f:R→R:f(x)=x2g:R→R:g(x)=sinxh:R→R:h(x)=3x​To find (h⋅(g⋅f))(x)…

Q: Suppose fn → f and the functions fn all satisfy the Lipschit condition |fn(x) – fn(y)| < Mx – y| for…

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