Question

Please prioritize part a)

1. (Integral Test Error estimate) (a) Let f : [1,00) + [0,00) be non-increasing, with so of converging. Let ak = f(k). Show for n in N that L., 540 ) at sex =Ě«-{«< [*60 a. f(t) dt < En:= k – ak < f(t) dt. Jn 1 k=1 Jn Determine for each series below, the minimal n (within 1) which ensures that E, < 10-6 mm ( min) which ensures

Answer 1

rom- in creasing Solution :- (a) let with cet f! [P,00) —> [0,0)) be aff Corverging. ak = f(KI Show : mia Eak- walk Kal K=] nti Jiya nh kanal i k To Show : S 114) df F Em = 20K - safada : 9pm dt = The data fl) is non pleca increasing frenetim 19 ) felt "Spez) olt" - fik) v Ken So, I stayed t affect a FA s i pe dt = { film e lirin) - f(0) - Fik) : = En so s fle) des En kant mu K=1 741

So, we know that agful) df - Ê 'freelle kom k ? s P1k41) dt (4). < perehe) P is non-moueating) $(kn) Kank So, ay Fle) olf - $ $14. df So, I fly at > Ë fue) := En So, I 9818alf or if I fiun) - no K=nt) Pure) s, J18974 k =) 7+1 167 ensures that within 1) which To find minimal on En < 10-6 8W8 then by previous part. ut f(x) = 1 1 1 we know En s s frt) alt 30, we en su pistolt < 10-6 7 Set of 106 > <104 7" E 10-6 IT

i 185% > 1006 1 02 > 50000 yn > ✓ 50000 223.60 so, minimal n = 223. 10) - Î by glometric scries Sum. K=1 and a bok = $11-07"). (1-19") 3, 4 - 1 (5 - 6 (3) So, En long deja < 1016 9 99 e 2 x 10-6 ay 37% 103 In > logo (o so, me gj u 1: 3"= 59049 >