Step-by-step explanation:
โโโโยฐยฐ (2x + 3)โฟ / (n9โฟ)
Use ratio test:
lim(nโโ)โaโโโ / aโโ< 1
lim(nโโ)โ[(2x + 3)โฟโบยน / ((n+1) 9โฟโบยน)] / [(2x + 3)โฟ / (n9โฟ)]โ< 1
lim(nโโ)โ(2x + 3) n / (9 (n+1))โ< 1
โ(2x + 3) / 9โ< 1
โ2x + 3โ< 9
-9 < 2x + 3 < 9
-12 < 2x < 6
-6 < x < 3
If x = -6, โโโโยฐยฐ (2(-6) + 3)โฟ / (n9โฟ) = โโโโยฐยฐ (-1)โฟ / n, which converges.
If x = 3, โโโโยฐยฐ (2(3) + 3)โฟ / (n9โฟ) = โโโโยฐยฐ 1 / n, which diverges.
The interval of convergence is therefore [6, 3).
โโโโยฐยฐ (x + 4)โฟ / n!
Use ratio test:
lim(nโโ)โaโโโ / aโโ< 1
lim(nโโ)โ[(x + 4)โฟโบยน / (n+1)!] / [(x + 4)โฟ / n!]โ< 1
lim(nโโ)โ(x + 4) n! / (n+1)!โ< 1
lim(nโโ)โ(x + 4) / (n + 1)โ< 1
0 < 1
The interval of convergence is therefore (-โ, โ).