setting up infinite sequence and plotting

How do I set up and plot:

Sum (n=0 to infinity) (n^3+1)

Thank you

[John D'Errico]

What have you tried? If nothing, then why not? Make an effort.

Of course, you may need to consider the series you show grows rapidly to infinity. So I might guess your question does not even show the series you actually want to form. Perhaps you mean the sum of 1/(n^3+1), not the sum of (n^3+1). That is convergent, apparently to:

vpa(ans)
ans =
1.6865033423386238859646052121869

Answer 1

[charan]

Hello Robert,

I am assuming that you want to sum 1/(n^3+1) and not n^3+1 since the sum for n^3+1 tends to infinity as n moves to infinty. The sum of 1/(n^3+1) converges to a value as n tends to infinity. You can try doing this by calculating the value of the sum for each n value in a "while" loop. You can store the previous and current sum values. Then you can decide a tolerance value within which if there is no change you can accept the sum as converged value. It can be done similar to the code shown:

old_sum=sym(0);
new_sum=sym(1);
data=new_sum;
n=1;
tol=0.000001;
while((new_sum-old_sum)>tol)
old_sum=new_sum;
new_sum=new_sum+(1/(n^3+1));
n=n+1;
data=[data,new_sum];
end
vpa(old_sum)

ans = 

1.6864528398592121974925758815338

vpa(new_sum)

ans = 

1.6864538398582121984926371784164

plot(0:n-1,data)

You can decrease the "tol" (tolerance) value for more accurate results. Here is the link to documentation of "vpa":

https://www.mathworks.com/help/symbolic/vpa.html

Answer 2

[tom]

1. Define the Sequence: Decide on the mathematical formula or rule that generates the sequence.
2. Generate Terms: Compute the terms of the sequence up to a reasonable number (since we can't truly generate an infinite number of terms).
3. Plot the Sequence: Visualize the terms using a plot.

For demonstration, I'll consider a few common types of sequences and plot them:

1. Arithmetic Sequence: A sequence in which each term after the first is obtained by adding a constant difference to the previous term. an=a+(n−1)da_n = a + (n-1)dan​=a+(n−1)dwhere aaa is the first term and ddd is the common difference.
2. Geometric Sequence: A sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio. an=arn−1a_n = ar^{n-1}an​=arn−1where aaa is the first term and rrr is the common ratio.
3. Harmonic Sequence: A sequence in which each term is the reciprocal of an arithmetic sequence. an=1na_n = \frac{1}{n}an​=n1​

Let's plot the first 50 terms of each of these sequences.Arithmetic Sequence

• First term a=1a = 1a=1
• Common difference d=2d = 2d=2

Geometric Sequence

• First term a=1a = 1a=1
• Common ratio r=2r = 2r=2

Harmonic Sequence

• an=1na_n = \frac{1}{n}an​=n1​

I'll generate and plot these sequences in Python.

Here are the plots for the three different types of sequences:

1. Arithmetic Sequence (an=1+(n−1)×2a_n = 1 + (n-1) \times 2an​=1+(n−1)×2): The terms increase linearly.
2. Geometric Sequence (an=1×2(n−1)a_n = 1 \times 2^{(n-1)}an​=1×2(n−1)): The terms increase exponentially.
3. Harmonic Sequence (an=1na_n = \frac{1}{n}an​=n1​): The terms decrease and approach zero as nnn increases.

Answer 3

[Prathamesh]

project in wed papge