How can I sketch a graph of the first two partial sums for the following power series:
Sum(n=1 to infinity) ((2x+5)^n) / (n(4^n))
just plot the graphs:
n=1: (2x+5)/4
n=2: (4x^2+36x+65)/32
http://www.wolframalpha.com/input/?i=plot+(2x%2B5)%2F4,+(4x%5E2%2B36x%2B65)%2F32
Notice they both cross the x-axis at x = -5/2
To sketch the graph of the first two partial sums for the power series, we will calculate the first two terms of the series and plot the points on a graph.
Step 1: Find the first term of the series
Plug in n = 1 into the series formula:
(2x+5)^1 / (1 * 4^1) = (2x+5) / 4
Step 2: Find the second term of the series
Plug in n = 2 into the series formula:
(2x+5)^2 / (2 * 4^2) = (2x+5)^2 / 32
Step 3: Calculate the first partial sum
Add the first two terms together:
First partial sum = (2x+5) / 4 + (2x+5)^2 / 32
Step 4: Calculate the second partial sum
Find the third term of the series by plugging in n = 3 into the series formula:
(2x+5)^3 / (3 * 4^3) = (2x+5)^3 / 96
After calculating the third term, add it to the first partial sum:
Second partial sum = (2x+5) / 4 + (2x+5)^2 / 32 + (2x+5)^3 / 96
Step 5: Plotting the points on a graph
Choose some x-values and substitute them into the partial sum formulas to find the corresponding y-values. Then, plot these points on a graph. Repeat this step to find more points to plot, if needed.
Once you have plotted a sufficient number of points, you can connect them to get a rough sketch of the graph of the first two partial sums of the power series.
To sketch the graph of the first two partial sums for the given power series, you need to calculate the values of the partial sums and plot them on a coordinate plane.
The first step is to find the formula for the nth partial sum (Sn). In this case, the nth partial sum is defined as:
Sn = Sum(k=1 to n) ((2x+5)^k) / (k(4^k))
To calculate the first partial sum (S1), we substitute n = 1 into the formula:
S1 = ((2x+5)^1) / (1(4^1)) = (2x+5)/4
Now, to calculate the second partial sum (S2), we substitute n = 2 into the formula:
S2 = ((2x+5)^1) / (1(4^1)) + ((2x+5)^2) / (2(4^2))
We simplify this expression to obtain S2 in terms of x:
S2 = (2x+5)/4 + (4x^2 + 20x + 25)/32
Next, we need to plot the points on a coordinate plane. Choose a few values of x and calculate the corresponding values of the partial sums using the formulas we derived earlier.
Let's choose x = -2, 0, and 2 as example values. Calculating the partial sums for these values will give us three points on the graph.
For x = -2:
S1 = (-4+5)/4 = 1/4
S2 = (-4+5)/4 + 4/32 = 1/4 + 1/8 = 3/8
For x = 0:
S1 = (0+5)/4 = 5/4
S2 = (0+5)/4 + 25/32 = 5/4 + 25/32 = 45/32
For x = 2:
S1 = (4+5)/4 = 9/4
S2 = (4+5)/4 + 81/32 = 9/4 + 81/32 = 117/32
Plot these three points on a coordinate plane and connect them with a smooth curve. This graph represents the first two partial sums of the given power series.