19 Power series[1K6]
All definitions and theorems needed to solve the following exercises may be found in Chap. 6 of [ 2 ] , Sec. 11.6 in [ 4 ] , or Chap. 8 of [ 22 ] .
- E406
[1K7] A power series \(∑_{k=0}^∞ a_ k x^ k\) has a positive convergence radius, if and only if, \(∃ ℓ {\gt}0\) for which \(|a_ k|≤ ℓ ^ k\) for every \(k≥ 1\).
Hidden solution: [UNACCESSIBLE UUID ’1K8’]
- E406
[1K9] Let \(c_ k\) be complex numbers, and \(a_ k=|c_ k|\). Note that power series \(∑_{k=0}^∞ a_ k z^ k\) and \(∑_{k=0}^∞ c_ k z^ k\) have the same radius of convergence \(R\).
Setting, for \(t{\gt}0\) real, \(\tilde f(t)=∑_{k=0}^∞ a_ k t^ k\), note that this formula defines a monotonic function \(\tilde f:[0,∞)→ [0,∞]\); show that the radius of convergence \(R\) coincides with the upper bound of \(t≥ 0\) such that \(\tilde f(t){\lt}∞\).
Hidden solution: [UNACCESSIBLE UUID ’1KB’] Hidden solution: [UNACCESSIBLE UUID ’1KC’]
- E406
[1KD]Prerequisites:3.Given \( f(t)=∑_{k=0}^∞ a_ k t^ k\) with \(a_ k≥ 0\), such that the radius of convergence is \(r{\gt}0\), show that \(\lim _{t→ r-}f(t)=f(r)\). Hidden solution: [UNACCESSIBLE UUID ’1KF’]
- E406
[1KG]Find two examples of series \( f(t)=∑_{k=0}^∞ a_ k t^ k\) with \(a_ k{\gt}0\) and with radius of convergence \(r\) positive and finite, such that
\(f(r){\lt}∞\)
\(f(r)=∞\)
Hidden solution: [UNACCESSIBLE UUID ’1KH’]
- E406
[1KJ]Find an example of a series \(f(t)=∑_{k=0}^∞ a_ k t^ k\) with \(a_ k∈ℝ\) and with radius of convergence \(r\) positive and finite, such that the limit \(\lim _{t→ r-} f(t)\) exists and is finite, but the series does not converge in \(t=r\).
Hidden solution: [UNACCESSIBLE UUID ’1KK’]
Note that (by Abel’s lemma) if the series converges in \(t=r\) then the limit \(\lim _{t→ r-} f(t)\) exists and \(\lim _{t→ r-} f(t)=f(r)\).
- E406
[1KM]Let \(b∈ℝ\), \(n∈ℕ\). Assuming that \(f(t)=∑_{k=0}^∞ a_ k t^ k\) with radius of convergence \(r\) positive and \(t∈(-r,r)\), determine the coefficients \(a_ k\) so as to satisfy the following differential equations.,
\(f'(t)=f(t)\) and \(f(0)=b\),
\(f'(t)=t^ 2 f(t)\) and \(f(0)=b\),
\(f''(t)=t^ 2 f(t)\) and \(f(0)=b,f'(0)=0\),
\(t f''(t) + f'(t) + t f(t)=0\) and \(f(0)=b,f'(0)=0\),
\(t^ 2 f''(t) + t f'(t) + (t^ 2-m^ 2)f(t)=0\) \(m≥ 2\) integer, \(f(0)=f'(0)=\ldots f^{(m-1)}=0\), and \(f^{(m)}=b\).
(The last two are called Bessel equations). [UNACCESSIBLE UUID ’1KN’]