Question

4. Let ($Q_n$) be a sequence of polynomials of degree $m_n$, and suppose that ($Q_n$) converges uniformly to $f$ on $[a, b]$, where $f$ is not a polynomial. Show that $m_n \to \infty$. 5. Use induction to show that $(1 + x)^n \ge 1 + nx$, for all $n = 1, 2, \dots$, whenever $x \ge -1$. Conclude that $(1 - t^2)^n \ge 1 - nt^2$ whenever $-1 \le t \le 1$.

          4. Let ($Q_n$) be a sequence of polynomials of degree $m_n$, and suppose that ($Q_n$) converges
uniformly to $f$ on $[a, b]$, where $f$ is not a polynomial. Show that $m_n \to \infty$.
5. Use induction to show that $(1 + x)^n \ge 1 + nx$, for all $n = 1, 2, \dots$, whenever $x \ge -1$.
Conclude that $(1 - t^2)^n \ge 1 - nt^2$ whenever $-1 \le t \le 1$.

4. Let (Qn) be a sequence of polynomials of degree mn, and suppose that (Qn) converges
uniformly to f on [a, b], where f is not a polynomial. Show that mn →∞.
5. Use induction to show that (1 + x)^n ≥ 1 + nx, for all n = 1, 2, …, whenever x ≥ -1.
Conclude that (1 - t^2)^n ≥ 1 - nt^2 whenever -1 ≤ t ≤ 1.

Added by Nancy S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

05/16/2022

Step 1

Step 1: **Base Case:** For n = 1, the inequality (1+x)n ≥1+nx becomes (1+x) ≥1+x, which is true for all x ≥ -1. Show more…

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4. Let (Qn) be a sequence of polynomials of degree mn, and suppose that (Qn) converges uniformly to f on [a,b], where f is not a polynomial. Show that mn→ ∞. 5. Use induction to show that (1+x)n ≥1+nx, for all n = 1,2,..., whenever x ≥ -1. Conclude that (1−t2)n≥1−nt2 whenever −1 ≤ t ≤ 1.