"I've believed" is correct and can be used in written English.
It is an example of a verb phrase, specifically the present perfect tense, which is used when referring to actions that have happened in the past, but have relevance to the present. For example, "I've believed in the power of kindness since I was a child."
Exact(56)
I've believed in what we've done.
That's something I've believed in strongly for a long time".
"I've never worked with someone I've believed in so much".
"I've believed this for a long time, and I still do," he said.
It fits quite well with what I've believed in my life.
"Why, sometimes I've believed as many as six impossible things before breakfast".
I've believed it since I can remember, I'm going to believe it until I die.
"I've believed for quite some time that Byron had passed on, that they had killed him.
For a long time, I've believed that if I focus on the preparation, then the results will come.
Those are the things I've believed in all my life, the things I've been standing up for.
Similar(1)
I remembered how I had believed him.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com