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 unbounded train. Σ f(n), where f(n) is a true unequivocal monotonically shapely creasing capacity that converges andn E Z+, s can be terminable, using unsuitable unimpaireds, as follows: dn <s Jk+1 Using the disproportion in (1) and 5, verify that Σ , E Θ(1). [10 points]

    Given an unbounded train s = sigma^eternity _n=1 f(n), where f(n) is a true unequivocal monotonically decreasing capacity that converges and n elementof Z^+, s can be terminable, using unsuitable unimpaireds, as follows: sigma^k _n=1 f(n) + unimpaired^eternity _k+1 f(n) dn lessthanorequalto s lessthanorequalto sigma^k _n=1 f(n) + unimpaired^eternity _k f(n) dn, k elementof Z^+ Using the disproportion in (1) and k = 5, verify that sigma^eternity _n=1 1/n^3 elementof theta (1).

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