Query 13

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GOOD STUDENT SOLUTION WITH COMMENT: For large values of n, this series is similar to 1 / 3^n. As n approaches infinity, 1 / 3^n approaches 0. A larger number in the denominator means that the value of the function will be smaller. So, (1 / 3^n) > (1 / (3^n + 1)). We know that since 1 / 3^n converges, so does 1 / (3^n + 1).

COMMENT: We know that 1 / (3^n) converges by the ratio test: The limit of a(n+1) / a(n) is (1 / 3^(n+1) ) / (1 / 3^n) = 1/3, so the series converges.

We can also determine this from an integral test. The integral of b^x, from x = 0 to infinity, converges whenever b is less than 1 (antiderivative is 1 / ln(b) * b^x; as x -> infinity the expression b^x approaches zero, as long as b < 1).

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*&*& The ratio test takes the limit as n -> infinity of a(n+1) / a(n). If the limit is less than 1 then the series converges in much the same way as a geometric series sum(r^n), with r equal to the limiting ratio.

In this case a(n+1) = 1 / (2n + 2)! and a(n) = 1 / (2n) ! so

a(n+1) / a(n)

= 1 / (2n+2) ! / [ 1 / (2n) ! ]

= (2n) ! / (2n + 2) !

= [ 2n * (2n-1) * (2n-2) * (2n - 3) * ... * 1 ] / [ (2n + 2) * (2n + 1) * (2n) * (2n - 1) * ... * 1 ]

= 1 / [ (2n+2) ( 2n+1) ].

As n -> infinity this result approaches zero. Thus the series converges for all values of n, and the radius of convergence is infinite. *&*&

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This is an alternating series with | a(n) | = .1^n, for n = 0, 1, 2, ... .

Thus limit{n->infinity}(a(n)) = 0.

An alternating series for which | a(n) | -> 0 is convergent.

sum(1/n^.999) diverges and sum(1 / n^1.001) converges, but doing partial sums on your calculator will never reveal this. The calculator is very limited in determining convergence or divergence.

However there is a pattern to the partial sums, which are 1, .9, .91, .909, .9091, .90909, ... . It's easy enough to show that the pattern continues, so the convergent value is .9090909... .

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** The general term is the coefficient of x^n.

In this case you would have the factors of x^2, x^3 etc. go down to p-1, p-2 etc.; the last factor is p minus one greater than the exponent of x. So the factor of x^n would go down to p - n + 1.

This factor would therefore be p ( p - 1) ( p - 2) ... ( p - n + 1).

This expression can be written as p ! / (p-n) !. All the terms of p ! after (p - n + 1) will divide out, leaving you the desired expression p ( p - 1) ( p - 2) ... ( p - n + 1).

The nth term is therefore (p ! / (p - n) !) / n! * x^n, of the form a(n) x^2 with a(n) = p ! / (n ! * (p - n) ! )

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The term

p(p-1) / 2 ! * x^2

is the n = 2 term, since x is raised to the power 2.

The term

p(p-1)(p-2) / 3 ! * x^3

is the n = 3 term.

What is different about the coefficients of these terms, and what the difference have to do with the power of n?

The first this we would notice is that the denominators for n = 2 and n = 3 are respectively 2 ! and 3 !.

Then we notice that the numerators p ( p - 1) and p ( p - 1 ) ( p - 2) for n = 2 and n = 3 consist respectively of 2 and 3 terms, with a clear pattern to the terms.

So we expect that the coefficients will continue as

p ( p - 1 ) ( p - 2 ) ( p - 3 ) / 4 !

p ( p - 1 ) ( p - 2 ) ( p - 3 ) ( p - 4 ) / 5 !

etc..

The denominator is equal to n !, as previously observed. The last number subtracted from p is one less than n, so the nth term would be

p ( p - 1 ) ( p - 2 ) * ... * ( p - (n-1) ) / n !

which can be written

p ( p - 1 ) ( p - 2 ) * ... * ( p - n + 1) ) / n ! .

p ( p - 1 ) ( p - 2 ) * ... * ( p - n + 1) )

is the same as

p ( p - 1 ) ( p - 2 ) * ... * ( p - n + 1) * (p - n) * (p - n - 1) * ... * 3 * 2 * 1 / ( (p - n) * (p - n - 1) * ... * 3 * 2 * 1 )

with the entire denominator matching the part of the numerator beyond the factor p - n + 1.

The numerator is still p ! and the denominator is (p - n) ! so

p ( p - 1 ) ( p - 2 ) * ... * ( p - n + 1) )

= p ( p - 1 ) ( p - 2 ) * ... * ( p - n + 1) * (p - n) * (p - n - 1) * ... * 3 * 2 * 1 / ( (p - n) * (p - n - 1) * ... * 3 * 2 * 1 )

= p ! / (p - n)!.

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Self-critique (if necessary):

I should add that the nth term is therefore (p! / (p-n)!) / n! *x^n, of the form a(n) x^2 with a(n) = p! / (n! *(p-n)!)

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Self-critique Rating: OK

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Question:

**** Query 9.5.18 (was 9.4.18). What is the radius of convergence of the series x / 3 + 2 x^2 / 5 + 3 x^2 / 7 + …?

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Your solution:

We have the coefficients a(1)= 1/3, a(2)= 2/5, and a(3)= 3/7, and we know that “n” will be the numerator for the nth term. The denominator is equal to 2n + 1. Therefore, the coefficient of the nth term is a(n)= n / (2n+1). We have to find the limit of | a(n+1) / a(n) | as n approaches infinity. The reciprocal of this limit will be our radius of convergence. We find that a(n+1) / a(n) = (n / (2n+1)) / (2(n+1) +1)) = (2n+3) / (2n+1)n / (n+1). When we divide by 2n, we get (2n+3) / (2n+1) = (1+3 / (2n)) / (1+1 / (2n)). As n approaches infinity, this approaches 1. The expression (2n+3) / (2n+1)n / (n+1) approaches 1^2, so the limit of a(n+1) / a(n) is 1. The radius of convergence is the reciprocal, so it is also 1.

confidence rating #$&*: 3

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Given Solution:

For this sequence we have the following coefficients:

a(1) = 1 / 3

a(2) = 2 / 5

a(3) = 3 / 7

The numerators for n = 1, 2, 3 are respectively 1, 2, 3, so the numerator of the nth term will be n.

The denominators for n = 1, 2, 3 are 3, 5 and 7. When n changes by 1 the denominator changes by 2, so one terms of the denominator will be 2 * n. The values of 2 * n are respectively 2, 4 and 6, which don't quite match the denominators 3, 5, 7. However if we add 1, we do get a match. So the denominator of the nth coefficient is 2 n + 1.

The coefficient of the nth term is thus

a(n) = n / (2 n + 1).

To find the radius of convergence you first find the limit of the ratio | a(n+1) / a(n) | as n -> infinity. The radius of convergence is the reciprocal of this limit.

a(n+1) / a(n) = ( n / (2n+1) ) / (n+1 / (2(n+1) + 1) ) = (2n + 3) / (2n + 1) * n / (n+1).

(2n + 3) / (2n + 1) = ( 1 + 3 / (2n) ) / (1 + 1 / (2n) ), obtained by dividing both numerator and denominator by 2n. In this form we see that as n -> infinity, this expression approaches ( 1 + 0) / ( 1 + 0) = 1.

Similarly n / (n+1) = 1 / (1 + 1/n), which also approaches 1.

Thus (2n + 3) / (2n + 1) * n / (n+1) approaches 1 * 1 = 1, and the limit of a(n+1) / a(n) is therefore 1.

The radius of convergence is the reciprocal of this ratio, which is 1.

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*&*& As seen in 9.4.6 we have

a(n) = p ! / (n ! * (p - n) ! ) so

a(n+1) = p ! / [ (n+1) ! * ( p - (n+1) ) ! ] and

a(n+1) / a(n) = { p ! / [ (n+1) ! * ( p - (n+1) ) ! ] } / {p ! / (n ! * (p - n) ! ) }

= (n ! * ( p - n) !) / {(n+1)! * (p - n - 1) ! }

= (p - n) / (n + 1).

This expression can be written as

(p / n - 1) / (1/n + 1). As n -> infinity both p/n and 1/n approach zero so our limit is -1 / 1 = -1.

Thus the limiting value of | a(n+1) / a(n) | is 1 and the radius of convergence is 1/1 = 1. *&*&

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&#Good responses. Let me know if you have questions. &#