Question

2.1: First, we need a little fact, but you may need to look up some things about In limits. Show that lim, O for any > 0. You will need L'Hôpital's rule. You may use the fact that lim, O for all > O and any constante. 2.2: So we can split the expression into something that looks weird: Some magie is about to happen, but first find the smallest inte ger value for where In() < (Use a calculator) 2.3: Notice that when Inn), we have that This means that 1. So then we can say that Σ. where is the answer from 2.2, since converents concerns the tail of a sexuence the first few terms don't alleet whether the sun is finite or not), we can always chop a few terms oil. Look at the series on the right side of the inequality, what kind of series is that? Does it converge or not? 2.4: What can you conclude about the convergence of

Answer 1

Soli 2 . lim ln (n) , pro x 400 HP line Eas). (Applying at the pitalitule Call lime Vue lim H → bekany 4400 bub

Now ln (n) ln (m) ln (m) 1/2 for mal, do (1)=0, bleu 3,4 I Fall ip. ln (n)<inle for h=2, In (2) = 0.693, 22. 1.414 -) In (n) <h2 *nl In (n) < n'z Illy Vnzi قا So In in 시 (47% = (uz ug F 12U A nly ha hla b3le 6% (u) ay * < T 3 * nol hal n2 h=) h/ Non are converger by potest Y So by composition test on the best Ž holn) also Converges. 24