Manipulating Known Series to Find New Series

The Big Idea

Many functions do not have a familiar power series that you can simply memorize. Fortunately, you usually don't need to start from scratch. Instead, you can begin with a known series and manipulate it until it becomes the series you want.

The most important known series is the geometric series:

This single formula is the starting point for a huge number of AP Calculus BC power-series problems. By replacing with another expression, multiplying the series by a constant or power of , differentiating term-by-term, or integrating term-by-term, we can generate many new series.

Four Ways to Manipulate a Known Series

1. Substitute an expression for the variable.
2. Multiply the series by a constant or a power of the variable.
3. Differentiate the series term-by-term.
4. Integrate the series term-by-term.

The key is to keep the original series visible while you manipulate it. That makes it much easier to see where every term comes from.

Method 1: Substitution

Suppose we start with the geometric series and replace with . Then:

That means we immediately know the series for . We did not need to calculate any derivatives of the function or discover the coefficients from scratch.

Method 2: Multiplication

Multiplying a known series by a power of shifts the powers of in every term. For example, if:

then multiplying both sides by gives:

This technique is especially useful when the function you need has an extra factor of or another power of the variable in the numerator.

Method 3: Differentiating a Series

One of the most powerful tricks is differentiating a known power series term-by-term. If:

then we can differentiate both sides:

Notice what happened: the exponent decreased by one, and the old exponent appeared as a coefficient.

Method 4: Integrating a Series

We can also integrate a known power series term-by-term. Starting with:

integrating gives:

The constant of integration is determined by the original function. For a function such as , evaluating at often makes the constant easy to determine.

Combining Multiple Manipulations

AP Calculus BC problems often require more than one manipulation. You might substitute first and then integrate, or multiply by a power of and then differentiate.

The safest strategy is to make one transformation at a time. Write the new equation after each step rather than trying to perform several algebraic operations simultaneously.

Changing the Center of the Series

A common AP Calculus BC task is to find a series centered at a value other than zero. The same manipulation ideas still work, but you should write the variable in terms of .

For example, if the known geometric series is expressed in powers of , then substituting an expression involving automatically produces a series centered at .

Always Track the Interval of Convergence

Manipulating a known series does not mean you can ignore convergence. The new series must converge on the appropriate interval, and substitutions can change the condition under which the original series converges.

For the geometric series, the fundamental condition is:

After making a substitution, solve the corresponding inequality for . Then check endpoints separately when the resulting interval includes them.

A Reliable AP Calculus BC Strategy

1. Identify a known series. The geometric series is often the starting point.
2. Rewrite the target function. Look for a form that resembles the known series.
3. Choose the manipulation. Substitute, multiply, differentiate, integrate, or combine these operations.
4. Transform one step at a time. Keep the algebra visible.
5. Rewrite the final result in standard power-series form.
6. Determine the interval or radius of convergence.

Common Mistakes

Practice Problems