Question

1. Consider a geometric series of \(\sum_{n=1}^{\infty} g r^{n-1}\). Plot the \(\mathrm{n}\) -th term \(a_{n}\), and the partial sum \(s_{n}\) versus \(n\) under the following two conditions.

i) with \(g=12\), and \(\mathrm{r}=2 / 3\),

ii) with \(g=12\), and \(r=3 / 2\).

Which of the (i) or (ii) converges? For the convergent one, approximate the total sum, i.e. \(\sum_{n=1}^{\infty} g r^{n-1}\) from your plot and draw a line on the plot to demonstrate this.

2. Using the following inequality, find the value of \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\) to within \(0.01\) of its exact value.

$$ S_{n}+\int_{n+1}^{\infty} f(x) d x \leq S \leq S_{n}+\int_{n}^{\infty} f(x) d x $$

Find the value of \(n\) through an iterative loop in MATLAB, and compute the lower bound, upper bound and the estimated value of sum.

Answer 1

`Hey,

Note: Brother if you have any queries related the answer please do comment. I would be very happy to resolve all your queries.

1)

a)

clc%clears screen
clear all%clears history
close all%closes all files
format long
g=12;
r=2/3;
a=[];
s=[];
for n=1:20
a(n)=g*r^(n-1);
if(n==1)
s(n)=a(n);
else
s(n)=s(n-1)+a(n);
end
end
plot(a);
title('Plot of terms');
figure;
plot(s);
title('Plot of sum of terms');

b)

clc%clears screen
clear all%clears history
close all%closes all files
format long
g=12;
r=3/2;
a=[];
s=[];
for n=1:20
a(n)=g*r^(n-1);
if(n==1)
s(n)=a(n);
else
s(n)=s(n-1)+a(n);
end
end
plot(a);
title('Plot of terms');
figure;
plot(s);
title('Plot of sum of terms');