(lim)┬(n→∞) (1^p+2^p+3^p+ …… +n^p)/n^(p+1) is equal to:a) 1/(p+1) b) 1/(p-1) c) 1/p-1/(p-1)

(lim)┬(n→∞) (1^p+2^p+3^p+ …… +n^p)/n^(p+1) is equal to a) 1/(p+1) b) 1/(p-1) c) 1/p-1/(p-1) d) 1/(p+2) Definite Integral Definition If an integral has upper and lower limits, it is called a Definite Integral. There are many definite integral formulas and properties. Definite Integral is the difference between the values of the integral at the specified upper and lower limit of the independent variable. It is represented as; ∫ab f(x) dx Properties of Definite Integrals Proofs Property 1: p∫q f(a) da = p∫q f(t) dt This is the simplest property as only a is to be substituted by t, and the desired result is obtained. Property 2: p∫q f(a) d(a) = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0 Suppose I = p∫q f(a) d(a) If f’ is the anti-derivative of f, then use the second fundamental theorem of calculus, to get I = f’(q)-f’(p) = – [f’(p) – f’(q)] = – q∫p(a)da. Also, if p = q, then I= f’(q)-f’(p) = f’(p) -f’(p) = 0. Hence, a∫af(a)da = 0. Property 3: p∫q f(a) d(a) = p∫r f(a) d(a) + r∫q f(a) d(a) If f’ is the anti-derivative of f, then use the second fundamental theorem of calculus, to get; p∫q f(a)da = f’(q)-f’(p)… (1) p∫rf(a)da = f’(r) – f’(p)… (2) r∫qf(a)da = f’(q) – f’(r) … (3) Let’s add equations (2) and (3), to get p∫r f(a)daf(a)da + r∫q f(a)daf(a)da = f’(r) – f’(p) + f’(q) = f’(q) – f’(p) = p∫q f(a)da Property 4: p∫q f(a) d(a) = p∫q f( p + q – a) d(a) Let, t = (p+q-a), or a = (p+q – t), so that dt = – da … (4) Also, note that when a = p, t = q...