Convergence of Series: Ratio and Root Tests

Introduction

• Purpose: Determine if a series converges using the ratio or root test.
• Tests to Use:
  • Ratio Test: Useful when terms have factorials or when the exponent involves n.
  • Root Test: Better when terms contain powers of n.

Example 1: Series ( \sum_{n=1}^{\infty} \frac{n}{3^n} )

• Initial comparison with geometric series ( \sum \frac{1}{3^n} ) failed as it converges.
• Ratio Test:
  • Calculate ( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ).
  • Evaluate: ( \frac{n+1}{3^{n+1}} \cdot \frac{3^n}{n} ).
  • Simplify: Limit simplifies to ( \frac{1}{3} ), which is less than 1.
• Conclusion: Series converges by ratio test.

Example 2: Series ( \sum_{n=1}^{\infty} \frac{2^n}{n!} )

• Ratio Test:
  • Use: ( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} ).
  • Evaluate: ( \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} ).
  • Simplify: Limit goes to 0.
• Conclusion: Series converges by ratio test.

Example 3: Series ( \sum_{n=1}^{\infty} \frac{(2n+1)^n}{n^{3n}} )

• Root Test:
  • Use: ( \lim_{n \to \infty} \sqrt[n]{a_n} ).
  • Evaluate: ( \left( \frac{(2n+1)^n}{n^{3n}} \right)^{1/n} ).
  • Simplify: Limit reduces to 0.
• Conclusion: Series converges by root test.

Example 4: Series ( \sum_{n=1}^{\infty} \frac{(-4)^n}{3^{n^2}} )

• Root Test:
  • Absolute value simplifies the alternating component.
  • Evaluate: ( \left( \frac{4^n}{3^{n^2}} \right)^{1/n} ).
  • Simplify: Limit goes to 0.
• Conclusion: Series converges by root test.

Example 5: Series ( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}} )

• Ratio Test:
  • Use: ( \frac{1}{\sqrt{(n+1)!}} \cdot \sqrt{n!} ).
  • Simplify: Limit goes to 0.
• Conclusion: Series converges by ratio test.

Example 6: Series ( \sum_{n=1}^{\infty} \frac{3^n}{n \cdot 2^n} )

• Ratio Test:
  • Evaluate: Limit results in ( \frac{3}{2} ).
• Conclusion: Series diverges because result is greater than 1.

Example 7: Series ( \sum_{n=1}^{\infty} \frac{n!}{n^n} )

• Ratio Test:
  • Use: ( \frac{n!}{n^n} ) and simplify.
  • Consider limit transformations and tricks (e.g., the exponential limit trick).
  • Simplify: Results in ( \frac{1}{e} ).
• Conclusion: Series converges by ratio test.

Example 8: Series ( \sum_{n=1}^{\infty} \left( \frac{1+2n}{3+4n} \right)^n )

• Root Test:
  • Use: Simplifies to ( \frac{1+2n}{3+4n} ).
  • Simplify: Limit goes to ( \frac{1}{2} ).
• Conclusion: Series converges by root test.

Conclusion

• Use the ratio test when dealing with factorials or exponential expressions in n.
• Use the root test when dealing with powers of terms.
• Always verify the limit conditions to determine convergence or divergence.