Homework Solution: Given an infinite series s = sigma^infinity _n=1 f(n), where f(n) is a continuous positive mon…

    8. Given an infinite series. Σ f(n), where f(n) is a continuous positive monotonically de- creasing function that converges andn E Z+, s can be bounded, using improper integrals, as follows: dn <s Jk+1 Using the inequality in (1) and 5, prove that Σ , E Θ(1). [10 points]
    Given an infinite series s = sigma^infinity _n=1 f(n), where f(n) is a continuous positive monotonically decreasing function that converges and n elementof Z^+, s can be bounded, using improper integrals, as follows: sigma^k _n=1 f(n) + integral^infinity _k+1 f(n) dn lessthanorequalto s lessthanorequalto sigma^k _n=1 f(n) + integral^infinity _k f(n) dn, k elementof Z^+ Using the inequality in (1) and k = 5, prove that sigma^infinity _n=1 1/n^3 elementof theta (1).

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    8. Given an unlimited succession. Σ f(n), where f(n) is a rectilineal decisive monotonically stereotyped creasing discharge that converges andn E Z+, s can be limited, using unbecoming undivideds, as follows: dn <s Jk+1 Using the dissimilarity in (1) and 5, ascertain that Σ , E Θ(1). [10 points]

    Given an unlimited succession s = sigma^eternity _n=1 f(n), where f(n) is a rectilineal decisive monotonically decreasing discharge that converges and n elementof Z^+, s can be limited, using unbecoming undivideds, as follows: sigma^k _n=1 f(n) + undivided^eternity _k+1 f(n) dn lessthanorequalto s lessthanorequalto sigma^k _n=1 f(n) + undivided^eternity _k f(n) dn, k elementof Z^+ Using the dissimilarity in (1) and k = 5, ascertain that sigma^eternity _n=1 1/n^3 elementof theta (1).

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