Answer by taking a quote from the following article:

Lee de Forest was born in 1873 in Council Bluffs, Iowa, the son of Anna Margaret (nee Robbins) and Henry Swift DeForest. He was a direct descendant of Jesse de Forest, the leader of a group of Walloon Huguenots who fled Europe in the 17th Century due to religious persecution. De Forest's father was a Congregational Church minister who hoped his son would also become a pastor. In 1879 the elder de Forest became president of the American Missionary Association's Talladega College in Talladega, Alabama, a school "open to all of either sex, without regard to sect, race, or color", and which educated primarily African-Americans.

One of de Forest's areas of research at Federal Telegraph was improving the reception of signals, and he came up with the idea of strengthening the audio frequency output from a grid Audion by feeding it into a second tube for additional amplification. He called this a "cascade amplifier", which eventually consisted of chaining together up to three Audions.  At this time the American Telephone and Telegraph Company was researching ways to amplify telephone signals to provide better long-distance service, and it was recognized that de Forest's device had potential as a telephone line repeater. In mid-1912 an associate, John Stone Stone, contacted AT&T to arrange for de Forest to demonstrate his invention. It was found that de Forest's "gassy" version of the Audion could not handle even the relatively low voltages used by telephone lines. (Due to the way he constructed the tubes, de Forest's Audions would cease to operate with too high a vacuum.) However, careful research by Dr. Harold D. Arnold and his team at AT&T's Western Electric subsidiary determined that by improving the tube's design, it could be more fully evacuated, and the high vacuum allowed it to successfully operate at telephone line voltages. With these changes the Audion evolved into a modern electron-discharge vacuum tube, using electron flows rather than ions. (Dr. Irving Langmuir at the General Electric Corporation made similar findings, and both he and Arnold attempted to patent the "high vacuum" construction, but the U.S. Supreme Court ruled in 1931 that this modification could not be patented).  After a delay of ten months, in July 1913 AT&T, through a third party who disguised his link to the telephone company, purchased the wire rights to seven Audion patents for $50,000. De Forest had hoped for a higher payment, but was again in bad financial shape and was unable to bargain for more. In 1915, AT&T used the innovation to conduct the first transcontinental telephone calls, in conjunction with the Panama-Pacific International Exposition at San Francisco.

was audio frequency amplification something he studied about?
One of de Forest's areas of research at Federal Telegraph was improving the reception of signals,