(a) Find the Maclaurin series for ln(1+x). (b) Use the alternating series estimation theorem to prove...
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(a) Find the Maclaurin series for ln(1+x). (b) Use the alternating series estimation theorem to prove that x-- -≤ln(1+x) ≤x. (c) Hence or otherwise prove that 2 x² 1 In(n + 1) − Inn < ¹. (d) Using your result in (c) show that 1+²+...+ -> ln(n+1). 2 n n (e) Hence deduce that the harmonic series 1 1 k diverges. (a) Find the Maclaurin series for ln(1+x). (b) Use the alternating series estimation theorem to prove that x-- -≤ln(1+x) ≤x. (c) Hence or otherwise prove that 2 x² 1 In(n + 1) − Inn < ¹. (d) Using your result in (c) show that 1+²+...+ -> ln(n+1). 2 n n (e) Hence deduce that the harmonic series 1 1 k diverges.
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