Calc1231W11T1 - Integral test for convergence

Sunday, 14 October 2018

3:51 PM

The integral test links the convergence of a sum to the convergence of an integral Let 
be an infinite series of positive terms. Suppose that f(x) is a continuous; positive and decreasing function where 
There are two cases to consider: 
ak is divergent 
i) If f(x) dx is divergent; then 
O 
f(k) = ak fork = 1, 2, 3, 
This is shown in the first figure because the area of the rectangles is greater than the 
1 
under the curve 
ii) If dc is convergent, then 
1 
under the curve 
(11 
convergent 
02 
2 
03 
3 
4 
5 
O 
infinite 
finite 
O area 
O area 
This is shown in the second figure because the area of the rectangles is less than the 
a2 
2 
03 
3 
4 
a5 
5 
ak is convergent But this is equivalent to the statement that 
ak is convergent because the difference between these two sums is 
Actually, the second diagram only shows that 
finite

 

We can use the integral test to determine if the series 
3 
converges. First we take a continuous, positive and decreasing function f : R —+ R where f(n) 
Now for any positive N 
dc 
f@) dc 
3 
for all n 
Such a function is 
1 
Then 
dc 
1 
So the series (1) is 
Note: the Maple syntax for oo is infinity. 
lim 
1/320 
1

 

We can use the integral test to determine if the series 
converges. First we take a continuous, positive and decreasing function f : R —+ R where f(n) 
xlsqrt(xA2+1) 
for all n 
Such a function is 
Now for any positive N 
dc 
1 
-sqrt(2)+sqrt(W2+1) 
Then 
dc 
1 
So the series (1) is 
Note: the Maple syntax for oo is infinity. 
lim 
dc 
1 
infinity

 

An important application of the integral test is to p-series. Recall that the sum 
converges precisely when p > 1 
We can check this fact by defining f@) 
continuous 
negative 
discontinuous 
positive 
Increasing 
decreasing 
Now, if p 1 then 
1 
n=l 
for x > 1 _ This function is (check all that apply): 
1 
Hence the convergence or divergence of the integral 
depends on p. 
• Ifp< 1 then I 
• Ifp> 1 then I 
infinity 
l/(p-l) 
1 
1 
O and the p-series 
and the p-series 
x dc 
diverges 
converges 
lim dc 
1 
O 
O 
by the integral test 
by the integral test 
• We analyse the remaining case p = 1 in the next question. 
Note: the Maple syntax for oo is infinity.

 

Now we consider the case of a p-series with p 
converges precisely when 
converges. By definition 
and so 
Hence the sum diverges by the integral test. 
1 According to the integral test 
1 
dc 
1 
1 
— dc 
1 
lim 
1 
Ill(N) 
1 
— dc 
1 
— dc 
infinity 
1

 

This is the famous harmonic series, whose divergence was first discovered by Nicole Oresme (1323-1382) 
His argument was that 
3 
4 
and since this series diverges, so does the harmonic series. 
2 
5 
8 
6 
8 
7 
8 
8 
8

 

 

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