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I need a check on the following:

Show that the sequence defined as $I_0=\log{\frac65}$, $$I_k + 5 I_{k-1} = \frac1k$$ $k=1,2,\ldots,n$ is not suitable to approximate, for $n$ large the value of the integral $\int_0^1 \frac{x^n}{x+5} dx$

Hint: start considering the perturbation of the initial data $\tilde I_0 =I_0 + \mu$

From the hint, I set $\tilde I_0 = I_0 + \mu $

By recursion, I find $$\tilde I_k = I_k + 5^k \mu$$ Therefore when $n$ is large, the true computed sequence $\{\tilde I_k \}_k$is far from the true one $\{ I_k\}_k$, and hence the results are really different.

Is it okay?

andereBen
  • 705

1 Answers1

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That is okay.

You can try that in Excel: Type

0  =LN(6/5)

in the first row, and

=A1+1 =1/A2-5*B1

in the second row, and extend to (say) 30 rows.

This is the data after a Excel 97/2003 computation:

k   I_k
0   0.182321557
1   0.088392216
2   0.05803892
3   0.043138734
4   0.03430633
5   0.028468352
6   0.024324906
7   0.021232615
8   0.018836924
9   0.01692649
10  0.01536755
11  0.014071341
12  0.01297663
13  0.012039925
14  0.011228946
15  0.010521935
16  0.009890325
17  0.009371907
18  0.008696021
19  0.009151473
20  0.004242637
21  0.026405862
22  -0.086574767
23  0.476352093
24  -2.340093801
25  11.740469
26  -58.66388348
27  293.3564544
28  -1466.746558
29  7333.767272
30  -36668.80303

It's clear from the data that $I_k$ is unreliable after $k\geq 22$.

LeechLattice
  • 326