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  3. assume that f is differentiable at a.compute limn?

Assume that f is differentiable at a.Compute limn??n?i=1k[f(a+in)?f(a)]Deduce the limit limn??n?i=1k[(n+i)mnm?1?1]

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Basic Math

Assume that f is differentiable at a.

  1. Compute limn??n?i=1k[f(a+in)?f(a)]

  2. Deduce the limit limn??n?i=1k[(n+i)mnm?1?1]

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3.7 Ratings (572 Votes)

  • Compute limn??n?i=1k[f(a+in)?f(a)]

    limn??n?i=1k[f(a+in)?f(a)]=?i=1k[limn??n[f(a+in)?f(a)]]

    Let t=in?n=it?i=nt
    ?i=1k[limn??n[f(a+in)?f(a)]]=?i=1klimt?0i?f(a+t)?f(a)t=?i=1kif?(a)=k(k+1)2?f?(a)

    Therefore, A=K(K+1)2?f?(a)

  • Deduce the limit limn??n?i=1k[(n+i)mnm?1?1]

    limn??n?i=1k[(n+i)mnm?1?1]=?i=1k[limn??n((n+i)mnm?1)]

    We have: limn??n?(n+i)m?nmnm=limn??n?[(n+in)m?1]=limn??n?[(1+in)m?1]

    Let u=in,n??,u?0
    limn??n?[(1+in)m?1]=limu?0iu[(1+u)m?1]=limu?0i[(1+u)m?1u],bylimu?0(1+u)m?1u=m??i=1kim=mk(k+1)2

    Therefore, limn??n?i=1k[(n+i)mnm?1?1]=mk(k+1)2

b).

Therefore, limn??n?i=1k[(n+i)mnm?1?1]=mk(k+1)2

a).

Therefore, A=K(K+1)2?f?(a)
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