4. The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that a, and bm are convergent series with an, b, 20 for all natural numbers n. Then, a,b,, is also a convergent series. (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 2 0 for all n and a, is convergent, then is also convergent. n=1 n=1 FACT 1: if a, 2 0 for all n and a, is convergent, then a < a, forn > M for some natural number M. (b) Byonly using the Comparison Test and the following fact (FACT 2), justify that if a,, b, 20 for all n and an: E b are convergent, then n is also convergent, where en is the maximum of an and bn. FACT 2: If a,, bn 2 0 for all n and 4 Ebn are convergent, then a,+ bn is n=1 n=1 also convergent. (c) By only using the Comparison Test and statements in (a) and (b), justify that if a, and bn are convergent and an, b, 2 0 for all natural numbers n, then n=1 > anb, is alsO convergent. 2.
4. The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that a, and bm are convergent series with an, b, 20 for all natural numbers n. Then, a,b,, is also a convergent series. (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 2 0 for all n and a, is convergent, then is also convergent. n=1 n=1 FACT 1: if a, 2 0 for all n and a, is convergent, then a < a, forn > M for some natural number M. (b) Byonly using the Comparison Test and the following fact (FACT 2), justify that if a,, b, 20 for all n and an: E b are convergent, then n is also convergent, where en is the maximum of an and bn. FACT 2: If a,, bn 2 0 for all n and 4 Ebn are convergent, then a,+ bn is n=1 n=1 also convergent. (c) By only using the Comparison Test and statements in (a) and (b), justify that if a, and bn are convergent and an, b, 2 0 for all natural numbers n, then n=1 > anb, is alsO convergent. 2.
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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Transcribed Image Text:4. The goal of this problem is to justify the following statement by using the Comparison Test
for series.
Suppose that a, and
bm are convergent series with an, b, 20 for all natural numbers
n. Then, a,b, is also a convergent series.
(a)
By only using the Comparison Test and the following fact (FACT 1), justify
2 0 for all a and , is convergent, then is also convergent.
that if
n=1
FACT 1: if a, 2 0 for all na and a, is convergent, then a < an for n 2 M for some
n=1
natural number M.
(b)
By only using the Comparison Test and the following fact (FACT 2), justify that
if an, be 20 for all a and an:bn are convergent, then Cn is also convergent,
n=1
where en is the maximum of an and ba.
FACT 2: If a,, b, 2 0 for all n and
bn are convergent, then an + b, is
n=1
n=1
also convergent.
(c)
By only using the Comparison Test and statements in (a) and (b), justify
that if a, and bn are convergent and an, b, 2 0 for all natural numbers n, then
n=1
> anb, is alsO convergent.
n=1
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