APM2611 Assignment
3 2024 - DUE 14
August 2024
QUESTIONS AND ANSWERS
[School]
[Course title]
,APM2611 Assignment 3 2024 - DUE 14 August 2024
Question 1
1. Find the radius and interval of convergence of the following series: (
i) ∞X n=1 100n n! (x + 7) n
(ii) ∞X k=1 (−1) k 10k (x − 5) k
2. Rewrite the expression below as a single power series: ∞X n=2 n(n −
1)cn x n + 2 ∞X n=2 n(n − 1)cn x n−2 + ∞X n=1 ncn x n .
Question 2
1. Verify by direct substitution that the given power series is a particular
solution of the DE (x + 1)y 00+ y 0 = 0 ; y = ∞X n=1 (−1) n+1 n x n .
2. Use the power series method to solve the initialvalue problem (x + 1)y 00
− (2 − x)y 0 + y = 0, y(0) = 2, y 0 (0) = −1; where c0 and c1 are given by the
initial conditions. 16 APM2611/101/0/2024
Question 3
Calculate the Laplace transform of the following function from first
principles: 1. f (t) = sin t if 0 ≤ t < π 0 if t ≥ π
2. f (t) = e −t sin t
3. Use Theorem 7.1 to find L{f (t)} (i) f (t) = −4t 2 + 16t + 9 (ii) f (t) = 4t 2 − 5
sin 3t (iii) f (t) = (e t − e −t ) 2
### Question 1
1. **Find the radius and interval of convergence of the following series:**
#### (i) \(\sum_{n=1}^{\infty} \frac{100^n}{n!} (x + 7)^n\)
To find the radius of convergence, we use the ratio test. Consider the
general term:
, \[a_n = \frac{100^n}{n!} (x + 7)^n\]
Applying the ratio test:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left|
\frac{100^{n+1}}{(n+1)!} (x + 7)^{n+1} \cdot \frac{n!}{100^n (x + 7)^n} \right|
\]
\[
= \lim_{n \to \infty} \left| \frac{100 (x + 7)}{n+1} \right|
\]
\[
= \left| 100 (x + 7) \right| \lim_{n \to \infty} \frac{1}{n+1} = 0
\]
Since the limit is zero for all \(x\), the series converges for all \(x\). Thus, the
radius of convergence \(R\) is infinite.
#### (ii) \(\sum_{k=1}^{\infty} (-1)^k \frac{10^k}{k} (x - 5)^k\)
Using the ratio test again:
\[
3 2024 - DUE 14
August 2024
QUESTIONS AND ANSWERS
[School]
[Course title]
,APM2611 Assignment 3 2024 - DUE 14 August 2024
Question 1
1. Find the radius and interval of convergence of the following series: (
i) ∞X n=1 100n n! (x + 7) n
(ii) ∞X k=1 (−1) k 10k (x − 5) k
2. Rewrite the expression below as a single power series: ∞X n=2 n(n −
1)cn x n + 2 ∞X n=2 n(n − 1)cn x n−2 + ∞X n=1 ncn x n .
Question 2
1. Verify by direct substitution that the given power series is a particular
solution of the DE (x + 1)y 00+ y 0 = 0 ; y = ∞X n=1 (−1) n+1 n x n .
2. Use the power series method to solve the initialvalue problem (x + 1)y 00
− (2 − x)y 0 + y = 0, y(0) = 2, y 0 (0) = −1; where c0 and c1 are given by the
initial conditions. 16 APM2611/101/0/2024
Question 3
Calculate the Laplace transform of the following function from first
principles: 1. f (t) = sin t if 0 ≤ t < π 0 if t ≥ π
2. f (t) = e −t sin t
3. Use Theorem 7.1 to find L{f (t)} (i) f (t) = −4t 2 + 16t + 9 (ii) f (t) = 4t 2 − 5
sin 3t (iii) f (t) = (e t − e −t ) 2
### Question 1
1. **Find the radius and interval of convergence of the following series:**
#### (i) \(\sum_{n=1}^{\infty} \frac{100^n}{n!} (x + 7)^n\)
To find the radius of convergence, we use the ratio test. Consider the
general term:
, \[a_n = \frac{100^n}{n!} (x + 7)^n\]
Applying the ratio test:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left|
\frac{100^{n+1}}{(n+1)!} (x + 7)^{n+1} \cdot \frac{n!}{100^n (x + 7)^n} \right|
\]
\[
= \lim_{n \to \infty} \left| \frac{100 (x + 7)}{n+1} \right|
\]
\[
= \left| 100 (x + 7) \right| \lim_{n \to \infty} \frac{1}{n+1} = 0
\]
Since the limit is zero for all \(x\), the series converges for all \(x\). Thus, the
radius of convergence \(R\) is infinite.
#### (ii) \(\sum_{k=1}^{\infty} (-1)^k \frac{10^k}{k} (x - 5)^k\)
Using the ratio test again:
\[
