Geometric Series and Partial Sums
The plot above illustrates both the terms of the geometric series (2/3)ⁿ and the sequence of its partial sums:
- The blue dots represent the terms of the series (2/3)ⁿ, which decrease towards zero as n increases.
- The orange dots represent the partial sums of the series, which converge to a specific value.
This visualization helps to understand how the individual terms of the series decrease, while the sum of the series approaches a finite value.
Formation of Partial Sums
Here are some calculations showing the formation of the partial sums for a few samples:
Sum for n=1: (2/3)¹ = 2/3
Sum for n=2: (2/3)¹ + (2/3)² = 2/3 + 2/3 = 4/3
Sum for n=3: (2/3)¹ + (2/3)² + (2/3)³ = 2/3 + 2/3 + 2/3 = 2
Sum for n=4: (2/3)¹ + (2/3)² + (2/3)³ + (2/3)⁴ = 2/3 + 2/3 + 2/3 + 2/3 = 8/3
Actual Sum of the Infinite Series
The sum of the infinite geometric series with a common ratio of 2/3 and the first term 1 is given by:
Sum = 1 / (1 – 2/3) = 3
As n increases, the sum of the partial sums approaches this value, demonstrating the convergence of the series.
Geometric Series and Partial Sums
The plot above illustrates both the terms of the geometric series (4/3)ⁿ and the sequence of its partial sums:
- The blue dots represent the terms of the series (4/3)ⁿ, which increase as n increases.
- The orange dots represent the partial sums of the series, which diverge, as the series does not converge.
This visualization helps to understand how the individual terms of the series increase without bound, and the sum of the series does not approach a finite value.
Formation of Partial Sums
Here are some calculations showing the formation of the partial sums for a few samples:
Sum for n=1: (4/3)¹ = 4/3
Sum for n=2: (4/3)¹ + (4/3)² = 4/3 + 16/9 = 28/9
Sum for n=3: (4/3)¹ + (4/3)² + (4/3)³ = 4/3 + 16/9 + 64/27 = 100/27
Sum for n=4: (4/3)¹ + (4/3)² + (4/3)³ + (4/3)⁴ = 4/3 + 16/9 + 64/27 + 256/81 = 388/81
Actual Sum of the Infinite Series
The series with a common ratio of 4/3 does not converge, so the sum of the infinite series does not exist.
Series and Partial Sums of \((-1)^n/n\)
Formation of Partial Sums
Here are some calculations showing the formation of the partial sums for a few samples:
Sum for n=1: \((-1)^1/1 = -1\)
Sum for n=2: \((-1)^1/1 + (-1)^2/2 = -1 + 1/2 = -1/2\)
Sum for n=3: \((-1)^1/1 + (-1)^2/2 + (-1)^3/3 = -1 + 1/2 – 1/3 = -5/6\)
Sum for n=4: \((-1)^1/1 + (-1)^2/2 + (-1)^3/3 + (-1)^4/4 = -1 + 1/2 – 1/3 + 1/4 = -7/12\)
…
This visualization helps to understand how the individual terms of the series oscillate, while the sum of the series does not converge to a finite value.
The blue dots represent the terms of the series \((-1)^n\), which alternate between -1 and 1 as \(n\) increases.
The orange dots represent the partial sums of the series, which alternate between -1 and 0 as \(n\) increases.
Partial Sums of the Series (-1)ⁿ/ⁿ
The following are the first 8 partial sums for the series (-1)ⁿ/ⁿ:
1. (-1)¹/1 = -1 ≈ -1.000
2. (-1)¹/1 + (-1)²/2 = -1 + 1/2 = -1/2 ≈ -0.500
3. (-1)¹/1 + (-1)²/2 + (-1)³/3 = -1 + 1/2 – 1/3 = -11/6 ≈ -1.833
4. (-1)¹/1 + (-1)²/2 + (-1)³/3 + (-1)⁴/4 = -1 + 1/2 – 1/3 + 1/4 = -7/12 ≈ -0.583
5. (-1)¹/1 + (-1)²/2 + (-1)³/3 + (-1)⁴/4 + (-1)⁵/5 = -1 + 1/2 – 1/3 + 1/4 – 1/5 = -37/60 ≈ -0.617
6. (-1)¹/1 + (-1)²/2 + (-1)³/3 + (-1)⁴/4 + (-1)⁵/5 + (-1)⁶/6 = -1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6 = -49/60 ≈ -0.817
7. (-1)¹/1 + (-1)²/2 + (-1)³/3 + (-1)⁴/4 + (-1)⁵/5 + (-1)⁶/6 + (-1)⁷/7 = -1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6 – 1/7 = -363/420 ≈ -0.864
8. (-1)¹/1 + (-1)²/2 + (-1)³/3 + (-1)⁴/4 + (-1)⁵/5 + (-1)⁶/6 + (-1)⁷/7 + (-1)⁸/8 = -1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6 – 1/7 + 1/8 = -67/105 ≈ -0.638
This series is known as the alternating harmonic series, and these calculations represent the partial sums up to the 8th term.
Terms of the Sequence \((-1)^n/n\)
The blue dots represent the terms of the sequence \((-1)^n/n\), showing the alternating pattern as \(n\) increases.
Partial Sums of the Sequence \((-1)^n/n\)
The orange dots represent the partial sums of the sequence \((-1)^n/n\), showing how the sum fluctuates as \( n \) increases.