We cover typical application problems involving repeating decimals, geometric applications, and the treatment of indeterminate forms.
Expressing Repeating Decimals as Fractions
0.272727… = 0.27 + 0.0027 + … = 0.27/(1−0.01) = 27/99 = 3/11
Example ①
Express the repeating decimal 0.4 (4 repeats indefinitely) as a fraction.
Solution:0.4/(1−0.1) = 0.4/0.9 = 4/9
Express the repeating decimal 0.4 (4 repeats indefinitely) as a fraction.
Solution:0.4/(1−0.1) = 0.4/0.9 = 4/9
Geometric Figures and Infinite Geometric Series
Connect the midpoints of the sides of square S_1 (side length 1) to create S_2, and continue in the same way. The area ratio is 1/2.
Sum of the areas = 1/(1−1/2) = 2
Example ②
Starting with equilateral triangle T_1 of side length 1, connect the midpoints of its sides to create T_2, and continue. Find the sum of the areas.
Solution:The area of T_1 is √3/4, and the area ratio is 1/4. Sum = (√3/4)/(1−1/4) = (√3/4)·(4/3) = √3/3
Starting with equilateral triangle T_1 of side length 1, connect the midpoints of its sides to create T_2, and continue. Find the sum of the areas.
Solution:The area of T_1 is √3/4, and the area ratio is 1/4. Sum = (√3/4)/(1−1/4) = (√3/4)·(4/3) = √3/3
Comprehensive Problems of the ∞ − ∞ Form
Rationalization is the standard method.
lim_{n→∞} (√(n+1) − √n) = lim 1/(√(n+1)+√n) = 0
Key Points
- Convert repeating decimals into fractions using infinite geometric series
- In geometric figures, the total often forms a geometric sequence at each stage
- Resolve ∞ − ∞ through rationalization or factoring
- Use the squeeze theorem to bound the absolute value from above
Practice Problems
- Express the repeating decimal 0.123123… as a fraction.
- Find lim_{n→∞} (√(n²+3n) − n).
- An infinite geometric series with first term 1 and common ratio r converges and has sum 4. Find r.
Answers and Explanations
- Answer:41/333
Explanation: 0.123/(1−0.001) = 123/999 = 41/333. - Answer:3/2
Explanation: Rationalize. 3n/(√(n²+3n)+n). Dividing the numerator and denominator by n gives 3/(√(1+3/n)+1) → 3/2. - Answer:r = 3/4
Explanation: 1/(1−r) = 4 → 1−r = 1/4 → r = 3/4. This satisfies |r|<1.